Relativisation and cylindric algebras

Chapter 5

Relativisation and cylindric algebras Outline of chapter In chapter 3 we investigated the basic properties of proper relation algebras, relation algebras, and representations. Here we continue that investigation as follows: 9 We generalise from proper relation algebras so that the unit or top element is not required to be an equivalence relation. The definition of a representation is generalised to that of a relativised representation. Corresponding to this, we generalise relation algebras by weakening or dropping the associativity axiom, obtaining the classes NA, WA, and SA of non-associative, weakly associative, and semi-associative algebras. We examine their basic algebraic properties ('arithmetic'). This investigation will continue in chapters 7 and 13. 9 We briefly consider an alternative kind of representation of relation algebras, where the boolean operations + , - need not be respected. These representations are called 'weak' representations. We will discuss the 'weakly representable relation algebras' in greater depth in chapters 17 and 18. 9 We briefly introduce cylindric algebras. Since the focus of the book is on relation algebras, we only introduce those parts of the theory of cylindric algebras that will be needed later on. We give references to the extensive literature on them as we go. We also discuss some connections between relation algebras and cylindric algebras, a topic that will be developed further in chapter 13. 151

Chapter 5. Relativisation and cylindric algebras

152

5.1

Relativisation

Several times already we mentioned Monk's theorem that the class of representable relation algebras cannot be defined with only finitely many axioms. Thefinitisation problem seeks ways around this and other similar negative results. There has been a great deal of research into this problem. One approach has been to consider alterations to the similarity type: can we achieve finite axiomatisability if we add additional operators; what happens if we delete some of the operators? We will discuss some of this work in section 6.4. An important alternative approach is to alter the definition of a representation, by relativisation. This is our topic in this section.

5.1.1

Relativised representations

A representation is an isomorphism from a relation algebra to a proper relation algebra, and in this way, proper relation algebras provide 'classical' semantics for relation algebras. Recall the definition of a proper relation algebra P over the base B: it is a field of binary relations on B containing the identity relation on B and closed under the boolean operations, conversion, and composition - - defined by rls -- {(x,y) C n • B: ~z C n((x,z) E r A (z,y) E s)}. There is a unit 1, the largest relation in the algebra. The unit is not necessarily B x B, but lemma 3.4 showed using closure under the operations that it is an equivalence relation over B. In the relativised approach to representations, the semantics are generalised, or weakened: the unit is some fixed binary relation W (not necessarily an equivalence relation) over the base, and the operations are relativised to W. The hope is that with such non-classical representations we may make inroads into the finitisation problem, or at any rate deepen our understanding of relation algebras. We will present these relativised representations here as models of a first-order theory. Recall from definition 3.30 that a (classical) representation of an relation algebra A can be viewed as a model of the first-order theory TA. Now we generalise that definition. D E F I N I T I O N 5.1 Let A be a BAO of the signature L ~ of relation algebras (see definition 3.9; we call such an A a relation-type BAO). 1. Write L(A) for the first-order signature consisting of a binary relation symbol for each element of A.

2. RA is the L(A)-theory consisting of the following axioms:

5.1. Relativisation

153

Vxy[ 1'A (x, y) ~ x = y] Vxy[r(x, y) ~ s(x, y) V t(x, y)]

Vxy[1A(x, y)

~ (r(x, y) ~ -~s(x, y))] Vxy[r(x, y) ~ s(y,x)] Vxy[1A(x,y)---, (r(x,y) ~ 3Z(S(X,Z) At(z,y)))] 3xy r(x, y)

for each r,s,t c A with:

r=s+t r -- --S r--~ r=s;t

rr

3. A relativised representation of A is a model of RA. 4. The base of a relativised representation M of A is dom(M). The unit of M is the binary relation (1A) M = {(x,y) : M ~ 1A(x,y)} on dom(M).

5. A complete relativised representation of A is a model M of RA such that for all x,y c M and each set S of elements of A such that the supremum ~ S exists in A, we have M ~ (Y~S)(x, y) iff M ~ s(x, y) for some s E S. 6. Where no confusion is likely, we will write the elements 0 A, 1A, 1'A of A and s simply as 0, 1, 1'. Relativised representations generalise classical representations. Any model of

TA (definition 3.30) is a model of RA. A classical representation M of a BAO A of the type of relation algebras is a special case of a relativised representation where 1M is an equivalence relation. But not all relativised representations are classical representations. In the RA axiom for composition, for example, we cannot conclude from s(x,z) At(z,y) that (s;t)(x,y). The reason is that (x,y) might not be an 'edge' in the relativised representation: we may not have M ~ 1(x,y). We will see examples later.

General relativised representations In the most general kind of relativised representation, all operations would be relativised to 1. We will temporarily call such representations 'general relativised representations'. A general relativised representation of a relation-type BAO A would be a model of the theory Vxy( 1(x, y) --+ [1' (x, y) ~ x = y]) Vxy(1 (x,y) --, [(r + s)(x,y) ~ r(x,y) V s(x,y)]) Vxy(1 (x,y) --~ [(-r)(x,y) ~ -~r(x,y)]) Vxy(l (x,y) ~ [~(x,y) ~ l(y,x) A r(y,x)]) Vxy(1 (x,y) --~ [(r;s)(x,y) ~-, 3z(l(x,z)Ar(x,z)A 1(Z,y)As(z,y))]) 3xy[1 (x, y)At(x,y)]

154

Chapter 5. Relativisation and cylindric algebras

for all r,s,t C A with t -~ 0. It can be checked that every model of RA is a model of this more complicated theory, but the converse is false. We will take a few moments to explain why we did not define RA like this. First, observe that because all quantifiers in the above theory are relativised to the unit, without loss of generality we can restrict the domain and relations to the unit. Formally, if M is a general relativised representation of A then so is the f_,(A)-structure M ~ with domain {x c M: 3y C M(M ~ l(x,y) V l(y,x))} and defined by r M' - r M N 1M for all r c A. Then M' ~ Vxy(r(x,y) ~ l(x,y)) for all r c A. So the RA-axioms Vxy[(r + s)(x,y) ~ r(x,y) V s(x,y)]), Vxy(1 (x,y) [(r;s)(x,y) ~ 3z(r(x,z) As(z,y))]), and 3xyt(x,y) are valid in M', for all r,s,t E A with t -~ 0. So these axioms are (without loss of generality) valid in general relativised representations, and there was no need to (fully) relativise them in RA. The remaining difference from RA is in the axioms for identity and conversion, which in RA are not relativised to 1. M' may not validate these. It can be checked that M' validates the identity axiom of RA iff the unit 1M' is reflexive on dom(M'), while M' validates the conversion axioms of RA iff 1M' is symmetric. Let us consider the merits of general relativised representations as semantics for relation algebras. It turns out that any relation algebra has a general relativised representation (completeness), but not every relation-type BAO with such a representation is a relation algebra (no soundness). The relation algebra axioms R0, R2, R5, and R7 of definition 3.8 do hold over relation-type BAOs with general relativised representations, but the other relation algebra axioms may fail. For a general relativised representation M of a relation-type BAO A, if 1M is not reflexive over its range then axiom R3, a , l ' = a, is not valid in A, and if 1M is not symmetric then axiom R4, a - a, is not valid. Axiom R1, associativity of composition, and axiom R6 may also fail. So general relativised representations are too general for our purposes here. However, they are important in other contexts. A finite axiomatisation for the isomorphism class of relation-type BAOs with general relativised representations has been defined in [Kram91]. In this paper and in, e.g., [And88, Mar99b], the reader will find a thorough investigation into various kinds of relativisation.

General relativised representations with reflexive symmetric unit Let us now, bit by bit, impose restrictions on the unit of a general relativised representation of a relation-type BAO A, and observe that each restriction causes A to validate a larger set of axioms. If we insist that the unit is reflexive over its range, then the identity axiom (axiom R3) becomes valid in A. If we also insist that 1M is symmetric, then axioms R4 and R6 become true. Thus, if A has a general relativised representation whose unit is reflexive and symmetric then A validates all the axioms for relation algebra except perhaps the associativity axiom (axiom R I ).

5.1. Relativisation

155

There is a converse: if axioms R3 and R4 hold in A then the unit of any general relativised representation of A, if there is one, is reflexive on its range and symmetric. So, we have informally established that for any relation-type BAO A that has a general relativised representation, the following are equivalent: 9 A has a relativised representation in the sense of definition 5.1 (i.e., a model of RA), 9 A has a general relativised representation M whose unit is reflexive on dom(M) and symmetric, 9 axioms R0 and R2-R7 of definition 3.8 are valid in A. Thus, the effect of our failing to relativise the axioms for 1' and v in RA is to limit ourselves to relativised representations of relation-type BAOs satisfying all the relation algebra axioms except perhaps associativity. Such algebras are called 'non-associative algebras'. This book is about relation algebras and is not concerned with algebras failing axioms R3, R4, or R6. In many cases, however, we are dealing with algebras of the type of relation algebras that we know satisfy all the relation algebra axioms, except that we do not know that they are associative. Non-associative algebras provide a useful catch-all generalisation. Unless otherwise stated, we will use 'relativised representation' as in definition 5.1 from now on, as we have seen that this is the appropriate definition for non-associative algebras.

5.1.2

Non-associative algebras

We now introduce these formally. D E F I N I T I O N 5.2 [Madd82, definition 1.2] A non-associative algebra is a relation-type algebra (i.e., an algebra of the same signature as relation algebras) that obeys all the relation algebra axioms (see definition 3.8) except perhaps the associativity axiom R1. NA denotes the class of all non-associative algebras. Of course, the term 'non-associative algebra' does not entail that the algebra is not associative, merely that it need not be. We next prove that the unit of any relativised representation of a relation-type BAO A is reflexive and symmetric. This follows from the fact, important in its own right, that a relativised representation of A is a boolean representation of the boolean reduct bool(A). We also show a little more formally than before that A must be a non-associative algebra.

156

Chapter 5. Relativisation and cylindric algebras

L E M M A 5.3 Let A be a relation-type BAO (or more generally, a relation-type algebra whose boolean reduct is a boolean algebra). Suppose that A has a relativised representation M ~ RA. Then

1. As boolean algebras, we have (A,O,I,+,-)~({rM'r6A},O,

1M u \)

where, for r 6 A, r M denotes the interpretation of the binary relation symbol r as a binary relation on M (so r M C_ 2M). 2. A is a non-associative algebra. 3. The unit 1M is a reflexive and symmetric relation on M. Proof 1. The isomorphism is evidently r H r M. By the axioms for 4- and - in RA, ( r + s ) M = r M U s M and ( - r ) M - 1M\r M. Obviously, 1 H 1M. We check that 0 maps to 0. If x, y C M and M ~ 0(x, y), then by the +-axiom, M ~ 1(x, y), so by the axiom for - , M ~ ~O(x,y), a contradiction. So 0 M -- 0. Finally, we check that the map r ~ r M is one-to-one. We have just seen that this map is a boolean homomorphism, so it is enough to show that r M -r 0 if r ~- 0 in A. But there is an axiom of RA saying exactly this. 2. The axioms R2-R7 are easily verified. As an example, consider the axiom R7" 6 " ( - ( a ; b ) ) < - b for all a,b c A. By part 1, it suffices to check that

M ~ V x y ( f ; ( - ( a ' b ) ) ( x , y ) --~ -b(x,y)). Assume for contradiction that M ~ 6 ; ( - ( a " b))(x,y) A ~ ( - b ) ( x , y ) for some x,y C M. Since c~'(-(a;b)) < 1, part 1 yields M ~ l(x,y). The negation axiom of RA now yields M ~ b(x,y). Also, by the composition axiom of RA, we may choose z C M with M ~ 6(x,z) A ( - ( a ' b ) ) ( z , y ) . So as above, M ~ 1(z,y). Also, the 'converse' axiom of RA yields M ~ a(z,x). So M 1(z,y) Aa(z,x) Ab(x,y), whence M ~ (a;b)(z,y) by the composition axiom of RA. Since M ~ 1(z,y) A (-(a;b))(z,y), the negation axiom yields M -~(a "b)(z, y), a contradiction. 3. If x c M then M ~ l'(x,x) by the identity axiom; and as 1' < 1, we have M ~ 1(x,x) by part 1. So 1M is reflexive. For symmetry, if M ~ 1(x,y) then by the axiom for conversion in RA, M ~ T(y,x), so as i < 1, the axiom for + shows that M ~ 1(y,x), too, proving symmetry. []

5.1. Rela6visation

5.1.3

157

Weakly associative algebras

However, even using relativised representations in which the unit is symmetric and reflexive, we find that the axioms for non-associative algebras are sound but not complete: there are non-associative algebras that have no relativised representations of this type. This is because just as relativised representations enforce conditions on 1' and ~, they also enforce a little associativity of ';'. This amount of associativity is encapsulated in the following definition. D E F I N I T I O N 5.4 [Madd82, definition 1.2] A weakly associative algebra A is an algebra of the type of relation algebras that obeys all the Tarski axioms for relation algebra (see definition 3.8) except perhaps the associativity axiom R1 (i.e., it is a non-associative algebra), but satisfying instead the weak associativity law (WL) ((l'.x);1);l:(l'.x);(1;l) for all x E A. Of course, 1 ; 1 = l, so the right-hand side may be simplified. WA denotes the class of all weakly associative algebras. See exercise 5.1 (6) for an example of a weakly associative algebra that is not a relation algebra, and (e.g.) exercise 12.6(5) for a non-associative algebra that is not weakly associative. P R O P O S I T I O N 5.5 If a relation-type BAO A has a relativised representation then A c WA.

Proof. By lemma 5.3, an algebra with a relativised representation must belong to NA, so the proposition is proved by verifying that the weak associativity law holds for such an algebra. This is the task of exercise 5.1 (1). [] The converse of proposition 5.5 also holds: this was proved by Maddux in [Madd82, theorem 5.20], and we will prove it by games in theorem 7.5. Thus, WA is precisely the class of relation-type BAOs with relativised representations (with reflexive and symmetric unit).

5.1.4

Semi-associative algebras

A stronger associativity axiom is the semi-associative law: D E F I N I T I O N 5.6 [Madd82, definition 1.2] A semi-associative algebra A is a non-associative algebra that satisfies (x;1) ; 1 : x ; ( l ;1), and, again, the right-hand side may be simplified to x; 1. SA denotes the class of all semi-associative algebras.

158

Chapter 5. Relativisation and cylindric algebras

It was shown by Maddux in [Madd91 b, theorem 25] that given any term built from variables and 1 using only composition ';', and in which at least one 1 occurs, its brackets can be rearranged arbitrarily without changing its value in a semiassociative algebra under any given assignment to variables. So, for example, SA ~ (x; 1);(y;(z;v)) -- ((x;(1 ;y));z);v. SA characterises a slightly stronger 1 class of relativised representations called 3-square relativised representations, defined below. Do not confuse these with the square classical representations of definition 3.3. D E F I N I T I O N 5.7 Let A be a non-associative algebra and let M be a model of RA ~ i.e., a relativised representation of A.

9 A clique C of M is a subset of the domain of M such that for all x, y c C we have M ~ l(x,y). 9 Let n be a cardinal. M is said to be n-square if for all cliques C of M with ]C] < n, all x,y E C, and all a,b E A with M ~ (a;b)(x,y), there exists z c M such that Ct_J {z} is a clique and M ~ a(x,z) Ab(z,y). Any classical representation is an n-square relativised representation, for any n. The class of relation-type BAOs with a 3-square (respectively, 4-square) relativised representation turns out to be precisely SA (respectively, RA). We'll see how to prove these results in chapters 13, where n-square and other 'locally classical' relativised representations will be used to capture classes of algebras nearer and nearer to RRA. Finally, what happens when we insist that the unit of a relativised representation of a non-associative algebra is transitive? Then the unit is an equivalence relation and any such relativised representation is a disjoint union of classical representations. Thus, the class of non-associative algebras with such relativised representations is just RRA, and this class cannot be defined using finitely many axioms. (In fact, the class of relation-type BAOs with a general relativised representation with transitive unit (even if not reflexive or symmetric) is not finitely axiomatisable [Mar99b].)

5.1.5

Basic facts about NA, WA, SA

These will be needed later. Many of their analogues for relation algebras were proved in [ChiTar51, J6nTar52]. Clearly, we have NA _~ WA _~ SA _~ RA. Recall that RA is a canonical variety (theorem 3.16). So are the others:

THEOREM 5.8 [Madd82, theorem 4.2] The classes NA, WA, and SA are canonical varieties. 1See exercise 5.1(6)

5.1. Relativisation

159

Proof By the proof of theorem 3.16, NA can be defined by Sahlqvist equations. Since NA can be defined by equations, it is a variety. Furthermore, for any A c NA, the canonical extension A + must satisfy all the Sahlqvist equations that are true in A (by theorem 2.95), so A + satisfies all the equations defining NA and therefore must belong to NA. Hence, NA is a canonical variety. Similarly, each of the other classes can be defined by the equations defining NA plus, in each case, a single equation that does not involve negation. By the same argument, each of these classes is therefore a canonical variety. []

Many facts about relation algebras hold for non-associative algebras too. The following extended version of the Peircean law (PL), saying in some sense that consistency of a triple is independent of its orientation, is a very useful example. L E M M A 5.9 (PL) Let A c NA. I f a ; b . ~ :/: O, then b ; c . ~ ~: 0 and [~"gt . c :/: O. Proof The first part is proved by lemma 3.12, and the second by the relation algebra axioms R4, R6, and lemma 3.11. The lemmas did not require associativity. []

Note also that in any non-associative algebra, T - 1 and 1' - 1', by lemmas 3.11 and 3.17. The following lemma contains some handy 'arithmetic facts' about weakly associative algebras, and will be useful on several occasions later. See exercise 10 below for the converse of item 6 when we are dealing with a semi-associative algebra. L E M M A 5.10 (Maddux 1982) Let A be an atomic weakly associative algebra and let a E AtA. Define st(a) - a ;~. 1' and end(a) - ~ ;a. 1' (the start and end of a, respectively). Then f o r all a, b C At A, 1. st(a),end(a) c AtA, 2. st(a) V - st(a), st(a)" st(a) - st(a), and similar equations f o r end(a), 3. st(a);a - a, end(a);6 - 6, 4. if a < 1' then st(a) - end(a) - a,

5. st(a) - end(d) and end(a) - st(d), 6. if a" b r 0 then end(a) - st(b). Proof The following was originally proved in [Madd82, theorems 2.2, 3.5]. We use the Peircean law (PL, lemma 5.9) repeatedly in this lemma.

Chapter 5. Relativisation and cylindric algebras

160

1. First, st(a) - a ; 6 . 1 ' :/: 0, else by the Peircean law we would obtain 1' ; a . a 0 which is false (we use 1' - 1' here, for example). Let x,x t c A t A with x,x ~ < st(a). We must show that x - x ~. Since x < st(a), we have x . a ; 6 -r so by (PL), a..~;a ~ O. Since a is an atom, a < ~ ; a < 1' ;a - a. Thus, (5.1)

.~;a -- a,

and similarly, .r' ;a - a. Hence, x ~;(.~;a) - a, so by (PL) we obtain (.~ ;a) ;~. x ~ -r 0. As ,r is an atom, ( i f ; a ) ' 6 _> x ~. Thus, x ~ < (if;l)" 1 < if; 1, by the weak associativity law (WL). By complete additivity of composition, there is an atom y such that x ~ < ~;y. But then, using (PL) and because y is an atom, we get y < x ; x ~. Finally, using x,x ~ < 1', we see that y < x; 1'- 1' ;x ~ = x. ,r Since y is non-zero and x,x ~ are atoms, we deduce that x - x ~. Thus, st(a) is an atom; and similarly, end(a) is an atom too. 2. Easy exercise. 3. Follows from (2) and equation 5.1 above. 4. By (3), a - s t ( a ) ' a < st(a)" 1' - st(a), and as a, st(a) are atoms we have a - st(a). The proof that a - end(a) is similar. 5. Immediate from axiom R4 and the definitions of st(a), end(f). 6. Suppose a ;b r 0, and let z be an atom with z < a" b. Then z<_a;b

=r =~

~ _< b ;~, ~ <_ (st(b)" b);s 6 <_ s t ( b ) ' w _< w

:=>

~ - st(b) ;~

by (PL) by part (3) some atom w, by (WE) and st(b) _< l' since 6, w are atoms.

Proceeding in the same way,

=~ =~ =~ =~

end(a) end(a) end(a) end(a) end(a) end(a)

_< ~ ;a _< ( s t ( b ) ' ~ ) ' a < s t ( b ) ' v _< v - st(b);end(a) _< st(b) - st(b)

by the above some atom v, as above since e n d ( a ) , s t ( b ) are atoms since end(a) _< 1' since they are atoms.

The following l e m m a will be very useful when we come on to chapter 7, because it shows that there is a consistent way of labelling the edges of 3-graphs with atoms in such a way that a given product of atoms is realised.

5.1. Relativisation

161

L E M M A 5.11 [ M a d d 9 1 b , l e m m a 49] Let A be an atomic weakly associative algebra, and let a, b, c C At A with a <_ b;c. 1. Let x, y be objects such that x - y iff a < 1'. There is a function f " {x, y} • {x,y} ~ A t A s u c h t h a t f ( x , y ) - - a a n d f o r a l l i , j , k C {x,y} we have f ( i , i ) <_ 1' and f ( i , k ) < f ( i , j ) ; f ( j , k ) . 2. Let x, y, z be objects such that x - y iff a < 1', x - z iff b <_ 1' and z -- y iff c <_ 1'. There is a function g" { x , y , z } • { x , y , z } ~ A t A such that g ( x , y ) - a, g(x,z) -- b, g(z,y) - c, and f o r all i , j , k E { x , y , z } we have g(i,i) < 1' and g(i,k) < g(i, j) "g ( j , k ) . Proof We consider just the second part; the first part is a special case. We define g on the remaining pairs from { x , y , z } as follows, g ( x , x ) - st(a), g(z,z) - st(c), g(y,y) - end(a), g(y,x) - ~, g(z,x) - [~, and g(y,z) - ~. These are atoms, by lemmas 3.17 and 5.10. Now use lemmas 5.9 (the Peircean law) and 5.10 to check that g is well-defined and satisfies the consistency conditions for all i, j , k E {x,y,z}. [] D E F I N I T I O N 5.12 A non-associative algebra is said to be integral if it satisfies 2 x;y-O

~

(x--O)V(y--O).

L E M M A 5.13 [ M a d d 9 0 b , t h e o r e m 4] Let A c SA be non-degenerate. Then A is integral if and only if the identity l' is an atom. Proof. Since A is non-degenerate, we have 0 -r 1 - 1 9 1'. Since 1 ;0 - 0, we have l' > 0 . If l' is not an atom, there i s x C A w i t h 0 < x < l'. Then l ' - x : / : 0 , and x ; ( l ' - x) _< 1' ;(1' - x) - l' - x, but also x;( l' - x) _< x; l' - x. Therefore, x ; ( l ' - x ) _< x. ( l ' - x ) - 0, so A is not integral. This holds for any non-associative algebra. Conversely, suppose l' is an atom. Let x, y E A, x, y :/: 0. We must show that x ; y ~ O. From .f - l' ;~..f :/: 0, the Peircean law (PL, lemma 5.9) gives .f'x. l' :/: 0. Since l' is an atom, we see that l' _< .f;x. Hence y - 1' ;y <_ (~;x) ;y _< (~; l) ; l -$" l, the last equality using the semi-associative law. Therefore, .f; 1 -y -r 0, so by (PL), x ; y . 1 ~ 0, as required. []

2Some authors additionally require that an integral non-associative algebra be non-degenerate.

Chapter 5. Relativisation and cylindric algebras

162

Exercises 1. Let A be a relation-type BAO with a relativised representation. Prove that A satisfies the weak associativity law (see definition 5.4), and so is a weakly associative algebra. 2. Check that any relation-type BAO with a general relativised representation satisfies axioms R0, R2, R5, and R7 of definition 3.8. Find a relationtype BAO with a general relativised representation that does not satisfy axioms R3, R4, and R6. [Start with a general relativised representation and construct the algebra from it.] 3. [Madd85] Prove that a non-associative algebra is integral if and only if x; 1 = 1, for all non-zero x. 4. [J6nTar52] Prove that an integral non-associative algebra is simple. 5. Find an example of a simple representable relation algebra that is not integral. 6. Let A be the relation-type BAO with atoms l',a,b, defined by the following relativised representation M on the set {0, 1,2, 3 } (see figure 5. I). aM

bM

= =

{(0, 1),(1,0),(2,3),(3,2)}, {(0, 2), (2, 0), (1,3), (3, 1)}.

Show that A c WA \ SA. Show directly that M is not 3-square. Show that 2

a

3

Figure 5.1: The weakly associative algebra A

lemma 5.13 fails over weakly associative algebras. Show that A is simple and subdirectly irreducible. [A was discovered by McKinsey; see, e.g., [Madd85, Madd90b].] 7. Prove that NA is a conjugated variety. 8. Prove that SA is a discriminator variety. [It may help to look up [Madd91b, theorems 8, 13].]

5.2. Weakly representable relation algebras

163

9. Prove that WA is not a discriminator variety. [It may help to look up [Madd9 l b, theorem 29].] 10. Let .~ be an atomic semi-associative algebra and let a,b be atoms of A. Prove that if end(a) - st(b) then a ;b -r 0. (See lemma 5.10 for definitions of st(a), end(a)). 11. Let M be a relativised representation of the relation algebra A: i.e., a model of RA. Prove that M is a complete relativised representation iff for every x, y c M, if M ~ 1(x, y) then M ~ a(x, y) for some atom a of A. 12. [Madd82, definition 5.10, lemma 5.12.1 ]. An identity atom e in a weakly associative algebra is an atom such that e < 1'. Let h be a complete, relativised representation of the weakly associative algebra q4p. Prove that the domain of the relativised representation can be partitioned into {Se" e is an identity atom of '14,'} in such a way that for any atom a C At('I'lf) we have (x, y) c h(a) =r x C Sst(a ) A y E Send(a). 13. Show that the weak associative and semi-associative laws are simple equations in the sense of definition 2.77, find their first-order correspondents, and compare with the correspondent of the associativity axiom given in lemma 3.24. Discover why the semi-associative law is so-called. 14. [Madd82, theorem 2.2(3)]. Let A be a set, E C_A, and let ": A ~ A be any function. Let C be any ternary relation on A. Consider the complex algebra with domain go(A), identity E, conversion defined by ~ = {Y:s E S}, and composition defined by S ; T = {a C A : 3s C S, t C T((s,t,a) C C)}, for any S, T C_A (see definition 2.65). Prove that this algebra is a weakly associative algebra iff the following conditions are met, for all a,b,c E A: (a) If (a,b,c) E C then (a,c,b), (c,b,a) C C. (b) a = b iff there is e C E such that (a,e,b) C C. (c) If e C E and (e,a,a), (a,b,c) E C then (e,c,c) C C.

5.2

Weakly representable relation algebras

J6nsson [J6n59] considered reducts of relation algebras obtained by dropping the operations + , - . The notion of representation for such algebras is weaker than for relation algebras, since representations need not respect the missing operations. J6nsson called the ones representable in this way 'weakly representable relation algebras'.

164

Chapter 5. Relativisation and cylindric algebras

Algebras of relations J6nsson's paper starts off by considering algebras closed only under the operations -, l',", ,. Roughly, he gave the name 'algebras of relations '3 to algebras of the form A - (A, I,N,-1,1) where A is a set of binary relations on some set X, the relational operations N,-1, and I have their usual meanings (-1 is conversion and I is relational composition), and I E A satisfies a l l -- a for all a E A. Such algebras need not be BAOs and need not even have interpretations of + , - , 0, 1 at all. Although I is not required to be the identity relation (equality) on X, J6nsson nonetheless proves [J6n59, p. 450] that any such A is isomorphic to an algebra B -(B, I , n , - 1 , I d x ) of binary relations on some set X; here, Idx denotes the identity relation {(x,x) : x E X} on X. Moreover, the proof shows that if 0 c A then the constructed isomorphism from A to B takes 0 to 0. (See exercise 2 below.) J6nsson [J6n59, theorem l] gave an axiomatisation of the class of all algebras isomorphic to some algebra of relations, using a method that we view as rather game-theoretic: see exercise 7.6(7). His axioms consist of nine basic equations together with an infinite set ~ of quasi-equations. He also gave a similar characterisation of the lattices of commuting equivalence relations. Weakly representable relation algebras Moving to relation algebras now, J6nsson [J6n59, p. 459] defines a relation algebra to be weakly representable if it is isomorphic to a relation algebra B whose elements are binary relations on some set, and in which the operations 0,-, ;, and ~ have their usual relation-operational meanings. Again, 1' need not be the identity relation on the base of B. However, since x; 1' - x is an axiom (R3) of relation algebras, the argument mentioned above shows that any relation algebra that is weakly representable in this sense is isomorphic to a relation algebra consisting of binary relations on some set X in which not only do 0,., ;, and ~ have their usual meanings but also in which 1' is Idx. This means that the following definition of weak representability of a relation algebra is equivalent to J6nsson's; it has been used in later papers such as [And94], and it is the definition we adopt in this book. D E F I N I T I O N 5.14 A relation algebra A is said to be weakly representable if there exists an isomorphism h : A ~ B, where B is a relation algebra, the domain of B consists of binary relations on some set X, and the operations 0,., 1', ~, and ; have their usual set-theoretic meanings: i.e., 0 B - 0, .B is intersection of relations (N), 1'B - Idx, ~B is relational conversion -1, and .B is relational composition 1. Such a map h is called a weak representation of A. We let w R R A denote the class of weakly representable relation algebras. 3We will only use the term 'algebra of relations' in this section, as in the context of this book it could well be confused with other notions.

5.2. Weakly representable relation algebras

165

We could define weak representations for non-associative algebras too, but it is easily checked that any weakly representable non-associative algebra is a relation algebra, so we do not need to do so. As with ordinary and relativised representability, we can view the notion of weak representability in terms of a first-order theory. Let A be a relation algebra. As in definition 3.30, the signature L(A) will consist of a binary relation symbol for each element of A. DEFINITION 5.15 Given a relation algebra A, the L(A)-theory WA is defined as follows:

Vx, y [l'A(x,y) ~-~ (x-- y)] Vx, y [R(x,y) ~ S(x,y) A T(x,y)] O',,(R,S) : Vx, y [e(x,y) ~ S(y,x)] o;(R,S, T) : Vx, y [e(x,y) ~ ~z(S(x,z) A T(z,y))] (YO : Vx, y ~OA(x,y) ~31-1 (R,S) : 3x, y [R(x,y) A~S(x,y)] G1, 9

o.(R,S, T):

for all R,S, T c A with R--S.T

R-~ R--S;T

R~S.

As usual, we will write simply 1' for 1'A, etc. It is easily seen that a relation algebra A is weakly representable if and only if WA has a model" indeed, any weak representation of A gives rise to a model of WA, and vice versa, just as for representable relation algebras. Note that for M ~ WA and (x,y) C 1M, the set {a c A" M ~ a(x,y)} is a filter of A, but not necessarily an ultrafilter. Results of J6nsson and Andr(~ka J6nsson remarked in [J6n59] that the nonrepresentable relation algebra constructed by Lyndon [Lyn50, p. 715] is not weakly representable. He then gave an example of a symmetric, integral relation algebra that is not weakly'representable. Strikingly, his construction involved nondesarguesian projective planeS. As we said in section 4.5, the projective plane construction of relation algebras was later modified by Lyndon [Lyn61]" Lyndon algebras were used by Monk in his proof of non-finite axiomatisability of RRA [Mon64]. The set E of quasi-equations mentioned above, combined with the equations defining RA, was shown to axiomatise wRRA in [J6n59, theorem 3]. Hence, wRRA is a quasi-variety. [J6n59, problem 1] asks whether Z can be replaced [in these results] by a finite set of axioms, or by a set of equations. Clearly, RRA c_ wRRA. J6nsson asked [J6n59, problem 3] whether wRRA RRA. Andr6ka answered this negatively in lAnd94]" indeed, she proved that RRA is not finitely axiomatisable over wRRA. wRRA was shown to be non-finitely axiomatisable implicitly in [Hai87, Hai91 ] and explicitly in [HodMik00]; the latter paper also proved analogous results for other classes of algebras. This provided a

Chapter 5. Relativisation and cylindric algebras

166

negative answer to the first part of J6nsson's problem 1 for wRRA and (since RA is finitely axiomatised) for the class of 'algebras of relations'. A result of J6nsson and Tarski A contrasting result was obtained in [J6nTar52]. J6nsson and Tarski consider a different kind of 'weak representation' - - this time, it is required that the following operations are respected: 0, l, +, l',", and ;. That is, all operations except perhaps the boolean complement and product ( - , .) must be interpreted in the natural, set-theoretic way. In [J6nTar52, theorem 4.22] they prove that every relation algebra has a representation of this type.

Looking ahead We will return to wRRA in exercise 7.6(7), where we describe J6nsson's axiomatisation method in terms of games, and also in chapter 17, where we prove non-finite axiomatisability of wRRA by approximately the method of [HodMik00]. In chapter 18, we will prove that it is undecidable whether a finite relation algebra is weakly representable, and indeed that there is no class K with RRA _C K c_ wRRA such that it is decidable whether a finite relation-type algebra is in K. However, wRRA is not so well understood as RRA, and some basic questions about it remain open as far as we know. For example, is it a variety? Is it canonical? Exercises 1. Show that McKenzie's algebra K (see section 4.4) is not weakly representable. 2. [J6n59] Show that if a relation algebra A has a representation respecting 9, 1', ~, and ;, then A is weakly representable (i.e., it has a representation respecting 0 as well). Show that if A has a representation respecting -,", and ;, then it is weakly representable. 3. Show that wRRA is a quasi-variety.

5.3

Cylindric algebras

So far, we have dealt with algebras corresponding to binary relations. A class of algebras corresponding to ~-ary relations, for any ordinal ~, is or-dimensional cylindric algebras. This subject, which was launched by Tarski and his students Louise Chin and Frederick Thompson as an algebraic counterpart to the semantics of first-order logic, is comprehensively dealt with in [HenMon+71, HenMon+85].

5.3. Cylindric algebras

167

An alternative approach, using polyadic algebras with substitution and permutation operators, was initiated by Halmos in [Hal62], and will be discussed a little in section 6.2. This book is focused on relation algebras and we do not propose to repeat the results of those volumes here, but we do give some of the elementary definitions and facts (often referring the reader to [HenMon+71 ] for proofs). This is for two reasons. First, it is generally acknowledged that results for relation algebras can in the main be replicated for cylindric algebras, though this is not always straightforward to do. Much of the work of parts II-V later can be carried through for cylindric algebras (we will refer the reader to published papers for the proofs). Second, the relationship between relation algebras and algebras of higher-dimensional relations forms an important subject later in this book (in chapters 13 and 15 especially), and we will need to know the basic properties of cylindric algebras in order to develop this material. We must distinguish two cases, roughly according to whether equality is in the signature or not: with equality we get cylindric algebra, without equality we get diagonal-free algebra (to be discussed later, in section 6.1). In first-order logic, a relation always has a finite arity. However in the algebraic version it turns out that it is unnecessary to impose this restriction. D E F I N I T I O N 5.16 [HenMon+71]

Let ct be an ordinal.

1. Recall that if U is a non-empty set, a U denotes the set of functions from ot to U. We write such functions as x, y, and for i < ct we write x(i) more concisely as xi. A subset of a U is called an ~-ary relation on U. For i, j < ct, the i, jth diagonal D i j denotes the set of all elements y of a U such that Yi -- Yj. Given i < t~ and an ot-ary relation X on U, we define the ith cylindrification CiX to be the set of all elements of a U that agree with some element of X on each coordinate except, perhaps, on the ith coordinate. That is, f i X = {y C aU : 3x E X Vj < c~(j 7~ i ~ y j -- xj) }.

2. A cylindric set algebra of dimension t~ is an algebra consisting of a set S of ot-ary relations on some base set U, equipped with the operations 0, 1 ( aU), U, \ (complement in aU), the diagonal elements Dij (i,j < o~), and the cylindrifications Ci (i < et). S must of course be closed under all these operations. The signature of or-dimensional cylindric set algebras consists of the boolean symbols 0, 1, +, - , constants dij (i, j < tx), and unary functions ci (i < ct). A structure for this signature is called a cylindric-type algebra, and a BAO of this signature is called a cylindric-type BAO. The class of all cylindric set algebras of dimension t~ is denoted Csa.

Chapter 5. Relativisation and cylindric algebras

168

3. A generalised cylindric set algebra of dimension a is a subdirect product of cylindric set algebras of dimension a. 4. A cylindric algebra of dimension a is defined to be an algebra C - - (C,O, 1, + , - , c i , d i j ) i , j < c L obeying the following axioms for every x,y c C, i,j,k < a: CO. (C, 0, 1, + , - ) is a boolean algebra CI. c i 0 - 0 C2. x <_ CiX C3. Ci(X" ciY) -- CiX" CiY C4. c i c j x - CjCiX C5. d i i -

1

C6. if k r i, j, then dij - ck(dik, dkj) C7. if i r j, then ci(dij, x). ci(dij. - x ) - O. These axioms are valid over cylindric set algebras. We write CAa for the class of all cylindric algebras of dimension a. 5. An a-dimensional cylindric algebra C is said to be representable if it is isomorphic to a generalised cylindric set algebra of dimension a; such an isomorphism is called a representation of C. RCAa denotes the class of all representable cylindric algebras of dimension a. Cylindric set algebras correspond, more or less, to square proper relation algebras. The cylindric algebraic counterpart of proper relation algebras is a subdirect product of cylindric set algebras. Every cylindric set algebra is a cylindric algebra (of the appropriate dimension), so every subdirect product of cylindric set algebras is a cylindric algebra, but in dimensions higher than l there are non-representable cylindric algebras. Most of the universal-algebraic results for relation algebras carry over to cylindric algebras. So CA~ is a conjugated variety of BAOs (fact 5.17 below), and hence is completely additive. The axioms are simple equations, hence Sahlqvist equations, so CAa is a Sahlqvist variety; by theorems 2.95 and 2.96 it is canonical and closed under completions. For finite (but not infinite) a, CAa is a discriminator variety: a discriminator term is c0c] ... ca_IX. See, e.g., [Madd91a] and exercise 3. It is known that RCAa is a variety [HenMon+85, corollary 3.1.108]. For a _< 2 it is finitely axiomatisable [HenMon+85, 3.2.56, 3.2.65], since RCAa - CAa for a < 1, and RCA2 - CA2 NMod{ci(x. y . c j ( x . - y ) ) <_ cj(cix. -d01) " {i,j} -- 2}.

169

5.3. Cylindric algebras

(The two formulas here are known as H e n k i n ' s two equations.) Monk proved that RCAa is not finitely axiomatisable for finite {x > 3 (see [Mon69], or [HenMon+85, theorem 4.1.3]). Many stronger negative results concerning axiomatisations of RCAa are known: see, e.g., [And97a, Ven97b]. We will be using several simple consequences of the cylindric algebra axioms, and we list the main ones below. The proofs can be done as exercises or can be found in [HenMon+71, chapter 1]. FACT 5.17 The following statements are valid in CAa for any ordinal ~ and any i , j , k < ~. 1. c i c i x -

cix

[HenMon+71, 1.2.3].

2. x . c,iy -- 0 iff y . CiX -- 0 [HenMon + 71, 1.2.5]. Hence,

C i

is conjugated and so

(by theorem 2.40) completely additive. 3. dij -- d j i [HenMon+71, 1.3.1 ]. 4. r

--

1 [HenMon+71, 1.3.2]. If k -r i,j then c k d i j -- dij [HenMon+71,

1.3.3]. 5. C-i(--CiX ) -- --C-iX

[HenMon+71, 1.2.111.

6. If x, y are atoms, then x <_ ciY iff c i x -- ciY (by items 1 and 2 above). 7. If x , y are atoms and x <_ cicjy, then C i C j X - CicjY, and there is an atom z with x < ciz and z < c.jy (by items 1, 2 above, axiom C4 and complete additivity of cylindrification).

Exercises 1. Calculate the Sahlqvist correspondents of the cylindric algebra axioms and of Henkin's two equations. (Cf. exercise 2.7(11).) 2. For n < 2, show that RCA,, is a conjugated Sahlqvist variety. Deduce that (a) RCA,, is canonical, (b) RCAn is atom-canonical and closed under completions, (c) the class StrRCAn of CA,z-atom structures whose complex algebras are in RCAn is elementary. ((2b) fails for finite n > 3 [Hodk97]. It seems likely that (2c) does too, but we do not currently have a proof.)

Chapter 5. Relativisation and cylindric algebras

170

3. For infinite t~ show that CAa is not a discriminator variety. Consider and investigate t~-dimensional cylindric algebras with an additional 'discriminator' operator: ca(x) - 0 if x = 0 and ca(x) = 1, otherwise. P R O B L E M 5.18 If C 6 RCAco does it follow that C + has a complete representation? For information on complete representations of o~-dimensional cylindric algebras, see [MonO0, section 15].

5.4

Substitutions in cylindric algebras

We will shortly investigate how to relativise a cylindric algebra and to form a relation algebra from a cylindric-type algebra. We will be using substitutions in cylindric algebras here and later in this book, so we begin with a short primer on them. For a fuller study, see [J6n62, HenMon+71, Res75, HenMon+85, Tho93], for example. Fix some dimension tx for our cylindric algebras, t~-dimensional cylindric algebras arise naturally from first-order logic restricted to t~ variables. Diagonals dij correspond to formulas xi = x j, and cylindrifications ci correspond to quantifiers ~xi. Now in first-order logic, we are all familiar with the process of substituting one free variable by another. Substitution can be achieved in cylindric algebra using the substitution operator s j,i defined as follows. D E F I N I T I O N 5.19 For i, j < or, and x any term of the signature of or-dimensional cylindric algebras, we define s~x-- { x, r

x),

The corresponding formula, if i -7/=j, is ~r

5.4.1

i f / - - j; otherwise. -- Xj A ~).

B a s i c facts a b o u t s u b s t i t u t i o n s

We will use the following facts about substitutions, taken from [HenMon+71]. FACT 5.20 Let ct be an ordinal and i, j , k , l < ct. The following are valid equations in CAa.

1. s~x <_ cix (by definition of sji and additivity of ci). 2.

sji (x" y)

- s~x. Sijr, S~.(--X) -- --S~X, sji (x -Jr-y) -- S~X + s~y. Indeed, whenever C c C Aa, X c_ C, and ,y_,X exists, then C ~ sji ~ , x - Y~xcx s~x. Also, in consequence, s ~ 0 - 0 and sji 1 - 1. That is, sji is completely additive and acts as a complete boolean endomorphism. [HenMon+71, 1.5.3].

5.4. Substitutions in cylindric algebras

171

3. If i r k, t h e n 5~dik -- djk. If i r k, l t h e n s~.dkl -- dkl. [HenMon+71, 1.5.4]. 4. djk. s~x -- djk-s~x [HenMon+71, 1.5.6].

5. S~CiX- CiX [HenMon+71, 1.5.8(i)]. 6. s~ckx- cks~x if k # i , j [HenMon+71, 1.5.8(ii)]. 7. cjs~x-- CiS[X [HenMon+71, 1.5.9(i)].

8. CiS~X- $~.X if i -r j [HenMon+71, 1.5.9(ii)]. 9. s~s~x - skx i if i r k [HenMon+71, 1.5.10(i)]. 10. s~s/kx- s~s~x if either i ~ {k,l} and k ~ {i,j}, or j - I. [HenMon+71, 1.5.10(iii,iv)]. 11.

i Jix SjS

__ S~X [HenMon+71, 1.5.10(v)].

12. SkSi x i J __ S~'S~X- s~s~x [HenMon+71, 1.5.10(ii,vi)]. These facts suffice for most elementary work with substitutions - - one uses them again and again. Thinking of s~x a s 3xi(xi - xj/~ ~,) and dij as xi - xj, the statements corresponding to these identities are easily seen to be valid. Or just think of the indices i < ot as variables xi, the operation sji as setting xi to (the contents of) xj ~ that is, 'xi : - xj' m and the operation ci as 'randomising' xi. In this light, the facts above seem intuitively natural. Considering fact 5.20(12), for example, we see that 9 ~Xi(Xi -- Xk/~ ~r

= Xi/~ ~ ) ) is logically equivalent to ~]Xj(Xj = Xk /~ 3Xi(Xi z Xj /~ ~) ),

9 setting xi to xk and then Xj to xi has the same effect as setting Xj to xk and then xi to xj. Similarly, randomising xi and then setting it to Xj is the same as setting it to xj straight off (if j -r i; cf. no. 8). However, the reader should beware: for finite ct, the CAa axioms are not strong enough to prove all logical equivalences between or-variable formulas, and not all 'natural facts' about substitutions are valid in cylindric algebras. For example, CAa ~ s2s~ 10 - - S 215052C2 x. Below, we will give additional conditions that are sufficient to ensure validity.

172

Chapter 5. Relativisation and cylindric algebras

5.4.2

M o r e valid s u b s t i t u t i o n - c y l i n d r i f i c a t i o n identities

The rest of this section is devoted to obtaining a more complete picture of which equations involving substitutions and cylindrifications are valid in CAm. This will be needed in chapter 13; we will also use it in theorem 5.44, though for that proof we could get by with just fact 5.20. For the rest of the section, fix an ordinal ct > 2. Unless otherwise stated, i, j,k,1, p , q denote arbitrary ordinals < ~.

s-c-words D E F I N I T I O N 5.21 1. An s-word is a finite string of substitutions (s~), a c-word is a finite string of cylindrifications (ck), and an s-c-word is a finite string of substitutions and cylindrifications, all of the signature of CAm. 2. We write ~ for the empty s-c-word. If u, w are s-c-words, we write simply uw for their concatenation9 The length of w (number of symbols ck, sji in w) is written JwI9 For example, ISoSlS2C2120 1 4. Clearly, for any s-c-word w and term t of the signature of {x-dimensional cylindric algebras, wt is also a term of this signature. And if x c C c CAm, then wx c C.

Program view of s-c-words Viewing sji a n d ck a s 'instructions' s e t X i t o Xj (i.e., X i "= X j ) a n d r a n d o m i s e Xk, respectively, we obtain a 'computational' view of any s-c-word: the xi (i < o~) are thought of as computer storage variables, and the word is thought of as a 'program' by reading it in this way as a sequence of instructions to be executed from left to right, one after the other. We mentioned this above as intuition for 2 o1S2C2 1 is: fact 5.20. As an example, the 'program' corresponding to SoS

begin X2 :-- Xo XO :-- Xl

X 1 := X 2

r a n d o m i s e x2 end After such a program has been run, starting with arbitrary values of the variables, the following possibilities arise for each xi: 9 The final value of Xi is the initial value of some Xj, this being forced by the program in the sense that it will always hold regardless of the initial values of the variables. For example, if i -r j then s~.cjs/k (or strictly, the program associated with it) ensures that the final value of Xk is the initial value of xj.

5.4. Substitutions in cylindric algebras

173

9 The final value of Xi is necessarily equal to the final value of some other variable xj. For example, cjs). ensures that the final values of xi and xj are equal. 9 The initial value of xi is destroyed during the execution of w (or equivalently of w'), by being overwritten or randomised. So there is no variable whose final value is forced to be the initial value of xi. For example, if i -~ j then both sji and s ji c j c i destroy the initial value of xi. Informally, we aim to prove, using fact 5.20 and a theorem of Thompson [Tho93], that whenever two s-c-words w, w~ satisfy, for all i, j < c~: 1. if w forces that the final value of xi is the initial value of x j, then so does w', and vice versa, 2. if w forces that the final value of xi is equal to the final value of x j, then so does w I, and vice versa, 3. the initial values of at least two variables xi, xj (some i < j < r stroyed during the execution of w (or equivalently of wl),

are de-

then w x - w~x is a valid equation in CAm. The first two conditions express that the non-deterministic programs associated with w, w' are 'equivalent'. The reader may check that they hold for each side of all equations of s-c-words in fact 5.20. For ~ > 4, the words and 52s~5~ satisfy all three conditions, so that CAm ~ Vx(s~s~ s~s~s~

50512052C2C31

Formalisation

We now proceed to formalise this 'program' view of s-c-words, using the approach of Resek-Thompson [ResTho91]. We will need some standard notation. D E F I N I T I O N 5.22 Define the substitution map [i/j] 9~ -~ ~ by"

[i/j](k)-

j,

if k - i ,

k,

otherwise,

where i, j < oL Of course, this definition depends implicitly on ~. We write maps on the left, and o denotes map composition, so that for example, ([1/2] o [2/3])(1) - [1/2]([2/3](1)) : 2. The identity map on a set X is as usual denoted by Idx. A possibly partial map g ' ( x ~ ~ is said to be finitary if {i < ~x" i r dom(g) or g(i) ~ i} is finite, ker(g), the kernel of g, denotes the binary relation on o~ (and equivalence relation on dom(g)) given by (i, j) E ker(g) iff i, j E dom(g) and g(i) - g(j). For r' c_ ~x, we write g-1 [1-'] for the set {i E dom(g) 9g(i) E F}. When F - {i} we write g-1 [i] instead of g-1 [{i}].

174

Chapter 5. Relativisation and cylindric algebras

DEFINITION

5.23

1. With each s-c-word w, we associate a partial finitary map ~ ' o ~ -~ o~ by induction on Iwl as follows" 9 ~-

Ida.

h

9 ws,j -

[i/jl.

9 wc"'/-- ~ o Ida\{/} -- wI~\{i}.

2. We define w* -

U ker(v~. Equivalently, w* - { (i, j ) " 3u, v ( w - uv A i, j C

W---U• dom(v~ A ~ ' ( i ) - ~'(j)) }. 3. For s-c-words w, w', we write A

9 w~w

1ifr

9 w ~ w' if w ~ w' and CAa ~ V x ( w x -

w'x).

Exercise 2 below shows that w* is an equivalence relation on ct. Of course, ~ , _~ depend implicitly on tx. The reader may wish to verify that w, w' satisfy the three 'program' conditions given above iff w ~ w' and Io~\ rng(w)l >- 2: see exercise 7 below. E X A M P L E 5.24 Let w - uv, where u is an s-word and v - ci0.., ci,_l a c-word. Let K - { i o , . . . , 6,- l }. Then w* - ker(~ U Idr). Hence, for two words w, w' of this form, we have w ~ w I iff ~ - w'. Equal s-c-words modulo CAa We will prove the following theorem" T H E O R E M 5.25 Let w, w' be s-c-words with w ~ w ~ and It~ \ rng(~)l >__2. Then CAa ~ Vx(wx-

w'x). Hence, w ~ w'.

This extends to s-c-words the following theorem of Thompson: FACT 5.26 [Tho93, theorem 3.6] Let ~ > 2 be an ordinal a n d let q, r < co. Assume that il, j l , . . . , iq, j q , k l , m l , . . . , k r , m r < Ct are such that [il/jl] ~ 1 7 6 [iq/jq] -[kl/ml]o . - . o [kr/mr] -- f C at~ and lot \ rng(f)l _ 2. Then CA~ ~

Vx(sS.tl... sjqiq (x) --

Sm " lk "l "

skrmr(X)).

Clearly, this is the restriction of theorem 5.25 to s-words. ([HenMon+85, 3.2.52] is a similar result, based on a semigroup theorem of J6nsson.) We will rely on it heavily in the proof.

5.4. S u b s t i t u t i o n s in cylindric algebras

175

Basic properties We begin with the following easy lemma which will be useful later. L E M M A 5.27 L e t u, w be s-c-words. 1. ff'~ -- ~o ~. 2. CAn ~ wdij - d~(i)~(j), f o r all i, j E dom(~). Proof.

1. We prove ff'~ -- fro ~ for all u, by a trivial induction on the length of w. If this is zero, then ~ - Ida and we are done. Assume the result for w, and let i, j < tx. Then uws) - f f ~ o [i/j] - (fro ~) o [i/j] (by the inductive hypothesis) A

-- ~ o ( ~ o [i/j]) - ~ o ws), as required. The proof for wci is similar.

2. The proof is by induction on the length of w. If this is zero, there is nothing to prove. Assume the result for u. We first prove it for w - us~. Let i, j G dom(r If i -- j, then in any m-dimensional cylindric algebra, we have dij - 1 - dff(i),~(j). So suppose that i r j. Now, by fact 5.20(3), uskdij -- Ud[k/l](i),[k/l](j); by assumption, [k/l](i), [ k / l ] ( j ) E dom(fi'), so by the induction hypothesis this i s dff([k/l](i)),~([k/l](j) ) --dr162 Next let w - uck. Assume that i , j E dom(~) - dom(u~ \ {k}. Then CAa ckdij -- dij, s o wdij - udij, which by the inductive hypothesis i s d f f ( i ) , f f ( j ) ; and this is clearly equal t o dr as required.

P R O P O S I T I O N 5.28 ~ is a c o n g r u e n c e on s-c-words: an e q u i v a l e n c e relation such that if u, u ~, w, w ~ are s - c - w o r d s a n d u ~ u t, w "~ w t, then uw ~ u~w ~. P r o o f ~ is clearly an equivalence relation. Assume that u ~ u' and w ~ w'. Then CAa ~ V x ( w x - w'x), so CAa ~ V x ( u w x - u'w'x). By lemma 5.27(1), ul"~ = ~ o ~ -- u' o w' - u'w'. So it remains to check that (uw)* - (u'w')*. Let ( i , j ) c (uw)* -- Uuw_xyker(y~. Pick s-c-words x , y such that u w - xy and (i, j) E ker(y~. If lyl < Iwl, then (i, j ) C w* - w'* C_ (u'w')*. Otherwise, y - zw for some z. By lemma 5.27(1), ~ ' - ~o ~. So (i,j) c ker(y~ implies (~(i), ~(j)) E ker(~. But z is a final segment of u, so ker(z~ C_ u* and (~(i), ~ ( j ) ) G u*. Since ~ - w' and u* - u'*, we obtain ( w ' ( i ) , w ' ( j ) ) c u'*, so clearly, (i,j) E (u'w')*. Thus, (uw)* C_ (dw')*; the converse is similar. []

The following is easily checked, using the definition of ~ and fact 5.20.

176

Chapter 5. Relativisation and cylindric algebras

LEMMA

1.

5.29

CiCj ~

CjCi

and t i C i ~ C i.

2. SjCi ~ C i 3.

s~ck "~ cks~

if k r i, j

4. cjs 5 ~ c/s / 5.

CiSj "-' Sji

6.

sjs~, ~ s~, if/-J=

7.

s~stk ~ s~s5 if either i ({ { k , l }

9

sSsS 9.

i

SkSi

J

ifiT~j k and k ~ { i , j } , or

- I.

i

,...., si _J

" "

k~k ~"~ S~Sj

DEFINITION s-c-word:

5.30 F o r any n < m and i o , . . . , i,,_ l < ~, we define the f o l l o w i n g

e,

P(io ....9 in-1 ) - LEMMA

j

Cio

if n - O, s{O

to"'Sio

in- 1

otherwise.

5.31 A s s u m e that I-" - { i o , . . . , in-1 } - { j o , . . . , j m - 1} C Or,f o r n, m < co.

Then: 1. P(i0 ..... i,-l) ~ P(J0 ..... Jm-l)"

2. If k c F, l < ~ then Proof

ck P(io ..... in-, ) "" s~p (it)..... in-

1) " "

P (it)..... in- 1 )"

B y the definition and l e m m a 5.29(6,7), w e have s ii "-' e, S ji S j i ~ S ji , a n d

ij.i k i k ij s io>io ~'~ Si0Si0.

So by p r o p o s i t i o n 5.28, we m a y a s s u m e that i o , . . . , t,,_ 1 are p a i r w i s e distinct, s i m i l a r l y for j o , . . . , jm-1, and n - m. If n _< 1, the result is clear, since P(io) - P(Jo)" A s s u m e that n > 1; let jo - ik, say. N o t e that by l e m m a 5.29(9,7), is " ~ s ; . ~" s i ki0, siki0 Sio

for all s < n .

(,)

5.4. Substitutions in cylindric algebras

177

So we have 9(io ..... in-l) =

io in- 1 Ci0 Si0 " " " Si0 . ..ik ..io tn- 1

Cio~io~io "'" S!o

by lemma 5.29(7)

..tO ..t l ln- 1 Cik:~(k :~i0 " ' " Si0 . t I io i2 tn- 1 Cik Sik Sik Si0 " " " Si0 il t2 tO i3 s!n-1 Ci k S!k Si k Si~ Si 0 9 to tl . tn- 1 Cik Si.k " " Si k $ti~ t1 S~.n- 1 St0 C j o S J . ' O ' ' " JO . JO sJOsJI gm-1 Cjo Jo J o ' ' ' S j o

io by lemma 5.29(4) and Sio ~ e

9

~---

,

9

"'" ~ ---

=

by(,)

,

by (,) again continuing ... by lemma 5.29(7) and s j~ J0 "~

E

9 ( j o ..... Jm-l)"

For the second part, if k c F then using the above and lemma 5.29(2), C k P ( i 0 ..... i n - l ) " " C k C k S k ~ n-I " ' ~ C k S k ~ n-I - - P ( i o ..... / n - l ) ' k io tn- 1 io tn- 1 sk P (io ..... in- l ) ~ S I c k Sk "'" Sk "" Ck Sk "'" Sk ~ P (io ..... in- 1)"

D E F I N I T I O N 5.32 We write Pr for any s-c-word of the form Ci0 S!0t o " " Si0in-~ w h e r e F - {io,..., i,,_ l} c_ ot for some n < 03. By lemma 5.31, Pv is well-defined up to _~-equivalence. We want to move cylindrifications rightwards within s-c-words, preserving ~_equivalence. The next lemma shows how to do this. Cf. [HenMon+85, theorem 3.2.51 (vi,vii)]. L E M M A 5.33 Let w be an s-c-word. Then

CiW ~

w0(~-l[i]).

Proof The proof is by induction on Iw[. We use proposition 5.28 and lemma 5.29 freely in the proof. If Iwl- o, there is nothing to prove. Assume the result for u, and let F = ff--l[i]. First we prove it for w - - ucj. Clearly, ~--1[i] = F \ {j}. Inductively, CiUCj ~'~ UpFCj. So we need only check that 13FCj ~ CjOF\{j }. If j ~ F, then cj commutes modulo ~ with every item in Pv, giving the result. If F = {j}, the result is trivial as cjs~cj ~ cj. If {j} c F, then take k c F \ {j} and observe that C j P F \ { j } ~ O F \ { j } C j ~'~ O F \ { j } S k CJ j ~'~ p F C j . Now we prove it for w - us~. Observe here that 1-4 clef 1~_ 1 [i] _ [j/k]_ 1[F] - (

Ft._Jr,\{j}, {j}'

if k E F, otherwise.

178

C h a p t e r 5.

Relativisation

and cylindric

algebras

Inductively, CiUS~ ~ UpFS~, SO we need to show that prs~ ~ s~pr,. 9

If k E F, then by the definition and lemma 5.31, prs~ ~ PF' ~ s~pr,. So assume that k ~ F. If also j ~ F, then s~ commutes with every term in Pr modulo ~ , so prs~ -~ s~pr - s~pF,. If F .

J J

.

.

{j}, then j ~ k, so lemma 5.29(5) gives prs~ =

.

cjsjs k ~ s~ _~ s~pF,. If {j} C F, then F has the form { j , m , l o , . . . , l n - 1 }, where j, m, 10,.. ., ln-] are distinct. Then SmS j kj ~ s~" and (modulo _~) s~ commutes with every entry in Pr,. So

prs~9

~

" ' ' S ml,,_l SJk" ,~ PI-'tS~ CmS/m0 . . . S mIn-lsJsJ""CmSlOm m k

~

s~pl ~'

as required.

[]

We can now prove theorem 5.25. Let w, w' be s-c-words with w ~ w' and let \ rng(~)[ _> 2. We show that by induction on the number n of cylindrifications ck (any k) in w w ' . If this is zero, so that w, w ~ are s-words, then the result follows from Thompson's theorem (fact 5.26). Let n > 0 and assume the result for smaller n. Without loss of generality, we have w - u c k v , where v is an s-word. Let K - ~ l[k]. By lemma 5.33,

Proof

w ~ w'

w-

UCk v ~ UVDK.

If K - 0, then lemma 5.33 gives w ~ ~ w '~ u v p r - - u v . Then w ~ ~ u v and u v w ~ has fewer ck than w w ~, so inductively, w ~ '~ u v ~ w , as required. Assume then that

K#0. Claim. w ~ also has the form u ~ck,v~, with K - v~

[U].

P r o o f of claim. Let i, j C K. Then i , j ~ d o m ( ~ ) - dom(w'), and since ~'(i) ~'(j) - k, we have (i, j) c w* - w'*. Therefore, we may write w ~ - u~xv ' where u', v' are s-c-words, x is an s-c-word of length 1, (i,j) E ker(v'), and Ivll is maximal

subject to this. Since v~(i) - v ~ ( j ) - k ~, say, by maximality of v~ and lemma 5.27(1) we must have k' ~ dom(~), so x - ck,. Thus, w t - u'ck, vt with ~/(i) - v'(j) - k'. Clearly, vt is the final segment of w ~ of maximal length such that i E dom(vt). This definition is independent of j. ~-.--|

Hence, fixing i and letting j range over K, we see that K C v~ [U]. The converse inclusion follows by running the same argument backwards. This proves the claim.

So by lemma 5.33, w ~ ~ ulV~pK. If K -- ~, then repeated use of lemma 5.31 gives "~ P r ~ u Iv~pK "~ w~, and we are done. Assume otherwise. Fix l ~ K, let

w "~ u v p r

K -- {k0,..., k,,-1 }, and define t - s ~ . . . s~"-'. We show that u v t ~ u ' v ' t .

5.4. Substitutions in cylindric algebras

179

1. uv"'t - u'ff~t. For, it is clear that ~ -

o

-

-

= Ida\K.

w", -

-

L e m m a 5.27(1) now gives o V / -

r

\K.

-

A

u'v't now follows from l e m m a 5.27(1), since rag(7) C_ o: \ K. 2. We check (uvt)* -- (u'v't)* as follows. Let i , j < tx. If (i, j) E (uvt)*, then since t is an s-word, (t(i),t'(j)) c (uv)*. Since t'(i) ~ K, we see that t'(i) -- ~'r(~(i)), and similarly for j. So (i'(i),~(j)) E (UVpK)* -- w* -- w'* = (u'v'pr)*. As before, this implies that (F(i),'i'(j)) c ( d r ' ) * , and it follows that (i, j) c (u'v't)*. The converse is similar. We have established that uvt ~ u~v;t. Also, uvtu;v;t has two fewer Cm than ww;. Inductively, we obtain uvt "~ u~v~t. Now repeated use of l e m m a 5.31 (as when K - ~x above) gives tOr ~ Or. We now obtain W ~

UVOK ~

uVtpK

~

Utv ttpr

~

u Iv ~9r ~ w t,

completing the proof.

[]

Exercises In the exercises, ~ is an ordinal and s-c-words are of the signature of CA~. 1. Check that the facts in 5.20 are valid in R C A a . Prove that some of the more interesting ones are valid in CAa. 2. Let w be any s-c-word. Show that w* is an equivalence relation on ~x. 3. Let w be an s-c-word such that ~ is one-one on its domain. Let o t \ d o m ( ~ ) c_ {ko,... ,k,,_l }, and v - WCko... Ck,_l. Show that v* - ker(Id~). 4. Prove that for any s-c-word w, any i < ~x with ~-1 [i] _ { i o , . . . , ik-1 }, and j C d o m ( ~ ) , we have S i~ ( j ) W ~ wz~j . ..it) . . . S ji k - I . Can you show S ~i ( j )

W ~

W S jio . . . S jik-I

too?

5. Show that the condition w* - w'* in theorem 5.25 is necessary" that is, show that ~ - w % and lot \ rng(~)] _> 2 do not imply that w ~ w'. [Try c0s I and

COC1 .] 6. There are special circumstances in which the condition w* - w ~* in theorem 5.25 is not necessary. An s-c-word w is said to be m o d e s t if for all s-c-words u, v and i < c~, if w - uciv then [?-- 1[i]1 <_ 1. For example, the concatenation of an s-word and a c-word is modest. Show that for modest w we have w* - k e r ( ~ U Idc~\dom(~)). Deduce that if w, w' are modest s-c-words A

and ~ -

w I then w ~ w t.

180

Chapter 5. Relativisation and cylindric algebras

7. Show that after running the 'program' associated with an s-c-word w, variable xi always ends up with the initial value of xj iff i c d o m ( r and r -j, and that variables xi, xj always end up equal iff (i, j) E w*. Deduce that s-c-words w, w' satisfy the three 'program' conditions of section 5.4.2 iff w ~ w' and [tx \ rng(~)[ _> 2.

5.5

Relativised cylindric algebras

Just as with relation algebras, we can generalise the notion of a cylindric set algebra by relativisation. The following definition is due to Henkin (cf., e.g., [Hen68]). D E F I N I T I O N 5.34 Let tx be an ordinal. 1. Let x, y be two t~-ary sequences of elements of some set, and let i < t~. We write x ~ i Y if for each j < ct, if j -r i then xj = yj. 2. A cylindric relativised set algebra o f d i m e n s i o n ct consists of a set S of txary relations over some base set U, equipped with constants 0 and 1 (here, the unit 1 is any t~-ary relation over U) and for i, j < or, the operations U, \ (complement relative to 1), diagonal elements Dij = {x C 1 : xi -- x j } , and the cylindrifications C i where Ci(X ) : {u C 1 : 3x C X, u ~i X}. S must contain the constants and be closed under the operations.

3. The class of all t~-dimensional cylindric relativised set algebras is denoted by Crsa. With relation algebras, we started with general relativisation and then imposed restrictions (symmetry, reflexivity) on the unit. We can do the same here with cylindric set algebras to obtain interesting classes of algebras. Fix an ordinal ct. Recall that the substitution map [i/j] : ct ~ ot is given by:

j, [i/j](k)-

k,

if k - i; otherwise.

D E F I N I T I O N 5.35 For any ordinal t~, 9 I)c~ is the class of all algebras in Crsa whose unit is closed under substitutions [i/j]: if x is in the unit and i, j < ct, then x o [i/j] -- (xo,. .. , X i - l , X j , X i + l , . . . )

is in the unit, too.

5.5. Relativised cylindric algebras

181

9 Ga is the class of all algebras in C r s a (and of Da) with unit closed under all maps" if rt- ct ~ ct is any map, and x is in the unit, then so is x o n (Xn(o),xn(1),...). Such units, or relativised representations with such a unit, are called 'locally square', or 'locally cubic'. For infinite ct, an intermediate class between Da and Ga can be obtained by requiring that if x is in the unit and n" tx ~ ct is finitary (i.e., it moves finitely many points), then x o n is in the unit. See, e.g., [Sai95]. L E M M A 5.36 Cylindrifications and substitutions are normal and completely additive in any algebra in Crsa (any tx). Proof Let C c Crsa. We show that cylindrifications are conjugated in C - just as in cylindric algebras. Let c,d C C and i < ct. Then c. cid 7g 0 iff there is Zt C c . c i d , iff there are a c c and b c d with a ~i b- This condition is symmetrical, so c. cid - 0 iff cic. d - O. By theorem 2.40, ci is normal and completely additive. Similarly, for distinct i, j < ct we have c. s~.d - 0 iff c . c i ( d i j , d ) - O, iff c i c . (dij " d) - O, iff (CiC . dij ) 9 d - O. So $ji is also conjugated, and so is normal and [] completely additive. The result for s i is trivial.

Of course, ~ S in the lemma will be different from U S if the latter is not an element of C. The question of when they coincide is related to complete representations (cf. section 2.3.8 for boolean algebras and definition 3.35 for relation algebras). We know that the class of completely representable boolean algebras coincides with the class of atomic boolean algebras (corollary 2.22). But we'll see later that the class of completely representable relation algebras is not elementary (theorem 17.6). Similarly, it can be shown that the class of completely representable or-dimensional cylindric algebras is not an elementary class [HirHod97a, theorem 34], for ct > 3. So this might surprise you: P R O P O S I T I O N 5.37 Let t~ be given and let C 6 Da be atomic. There is 99 E Da such that C ~- 9) and such that f o r any 6t 6 1D, there is an atom d 6 D with ~t 6 d. Hence, ~ D S -- U S f o r all S C 9) such that ~,D S exists. That is, any atomic C E Da has a complete relativised representation in the Da sense. By corollary 2.22, any algebra with such a representation must be atomic, so this result is best possible. A similar result can be proved for Crsa, but we will not use it. Proof If U is the base of C, let I - U At C c_ aU, and define a new C r s a 9) with unit I and domain {cN l ' c E C}. We check that 9) is indeed in C r s a (i.e., it is closed under the natural operations), that I is closed under substitutions so that ~D c Da, and that c H c N I is an isomorphism from C to D.

182

Chapter 5. Relativisation and cylindric algebras

By distributivity of operations over n in boolean algebras, it follows that D is closed under the boolean operations and that c ~ c N I preserves them. Clearly, {gt C l ' a i a j } - d C n l , so D contains the required diagonals and c H c n l preserves these. For cylindrifications, we let i < ~, x E C, and check that I N cCx -- {d E I" d ----i b for some b C I nx}. The inclusion ' 2 ' is clear. Conversely, assume that ,~ C I n c/Cx. Take an atom 7 of C with a c 7. Then a C 7" c/cx, so 7" c/cx -r 0 and 7 < c/cx- By lemma 5.36 and atomicity of C, there is 5 E At C with 8 < x and 7 < c/c& So a ~i b for some b c ~5. Since b c I n x and a E I, we are done. The foregoing shows that ~D is closed under the Crs operations and that c H c n I is a homomorphism from C onto D. This map is clearly one-one since it leaves atoms invariant. To show that ~D E Da, it suffices to show that D ~ sji l - 1 for alli, j < t x . (Cf. proposition 5.39(1) below.) This is clear, since C c Da so C ~ sji l - 1, and C-~ ~D. Since every ~ E ! is contained in some atom of D, theorem 2.21 shows that whenever S c D and ] ~ S exists, Y ~ S - US. [] The classes Crs, D, G have been very intensively studied (some references will be given at the end of the section). Fact 5.38 below records some information about them. The facts we quote will not be used heavily later on, but we feel it is worth including them for their intrinsic interest and because they foreshadowed developments in the theory of relation algebras to some degree - - for example, the theorem of Maddux ([Madd82], and theorem 7.5) that WA is the class of relation-type BAOs with relativised representations is perhaps the relation algebra analogue of fact 5.38. They also have wider significance. Consideration of Crs led to the decidable guarded fragment of first-order logic, which was introduced in [AndBen+98] and has generated an important field of research: see, e.g., [Ben96, Ben97, AreMon+99, Gr~i99a, Gr~i99b, HodOtt01] for more information. We will use the 'loosely guarded fragment' in chapter 19. This fragment has the finite model property, and this yields 'finite base property' results for Crs, D, G. Such results are not included in fact 5.38 since chapter 19 will cover this topic. Recall that for a class K of algebras, IK denotes the closure of K under isomorphism. FACT 5.38 Let ct > 3 be an ordinal. 1. CAa n ICrsa is a variety, axiomatised by the equations defining CAa, plus the two 'merry-go-round' axioms:

5.5. Relativised cylindric algebras

183

M G R I : SKS~CLX k~: = s ~ s ~ c ~ x for all distinct ~c,)~,p < M G R 2 : s~:s~s~sZckx ;~ ~ v - S~.v Sv~ S ~ c k x f o r all distinct lc, )~,la, v < ~.

Since CAa N ICrsa is axiomatised by simple equations, theorem 2.95 shows that it is a canonical variety. 2. ICrsa is a non-finitely axiomatisable conjugated canonical variety.

Its equational (and indeed, universal) theory is decidable. 3. IDa is a canonical variety, axiomatised by the axioms defining CAw with

C4 (CiCjX - CjCiX ) replaced by the weaker cicjx ~ cjcix, djk for k ~ i, j, plus MGR2. For finite ~, it is finitely axiomatisable and has decidable equational and universal theory. 4. IG2 - ID2 n M o d { x - d01 "( COC1(-d01" S0Clx" S~C0X)}, and for finite cz > 3, i j IGa-IOanMod{x<_cicj(s~cjx'sJcix'H 5ik SjSkCkX) k
i,j
" ir j

Hence, I G a is a finitely axiomatisable canonical variety. Its universal theory is decidable. For arbitrary cz, its equational theory is decidable. We now make some remarks on these facts. 1. The line of research of items 1-4 began with Resek, who showed in IRes75], inter alia, that CAa n I C r s a is axiomatised by the cylindric algebra axioms plus the so-called 'merry-go-round' axioms:

MGR

..~. ~Kl _~0 " :~K0

" "

-~"-2s~"-' czx -----s ;~ ~:o . . . SEn1 Kn- 1s ~ - ' SKI

_~:,-3_~,-2 ~Kn_2 5~ C~X~

for any 2 < n < co, ~c E n0~,a one-one sequence, and )~ < c~, ~, ~ rng(~c). Thompson simplified Resek's axioms by showing that all MGR axioms are implied by the instances MGRI and MGR2; this showed that CAa N ICrsa is finitely axiomatisable for finite c~. See [AndTho88] and [HenMon+85, 3.2.88] for more information. We note that MGR1 holds in a cylindric algebra with ~c - 0,/J - 1, )~ - 2 iff its relation algebra reduct (see section 5.6 below) satisfies x - x. 2. N6meti proved that ICrsa for or > 2 is a variety [N6m81]; a proof is also given in [HenMon+85, 5.5.10]. A construction of Andr6ka-N6meti shows that I C r s a is not finitely axiomatisable for cz > 3 [HenMon+85, 5.5.12]. Thompson adapted Resek's equations to obtain an explicit equational axiomatisation of ICrsa (see [ResTho91, p. 520]); a different proof using a

184

Chapter 5. Relativisation and cylindric algebras step-by-step technique of Andr6ka can be found in [Mon93] and the updated [Mon00]. The equations required are a variant of MGR together with some cylindric algebra axioms (excluding C4) and simple consequences of them. Canonicity of ICrsa was proved by Goldblatt in [Go195, theorem 4.6]. For conjugation see lemma 5.36. The decidability of the equational theory of Crsa for cz < co was proved by N6meti in [NEm86] and also in [NEm95, theorem 4.3]. Decidability of the universal theory of Crsa for all cz, plus other results, are established in [N6m96, theorem 4.1, remark 4.14]. See [AndGol+98] and [N6m95, w

[N6m96] for more information.

3. The axiomatisation of IDa was established by Thompson, adapting Resek's axiomatisation of CAa n ICrsa. A short proof, by the technique of Andr6ka mentioned above, can be found in [AndTho88]. The axioms are Sahlqvist equations, and so IDa is canonical. Decidability of the equational and universal theories of Da for finite cz was proved by N6meti in [N6m86, theorem 10(ii)] and [N6m95, theorem 4.31, and in [N6m96, theorem 4.11, respectively. [AndTho88, p. 681], [N6m95, remark 4.151, and [N6m96, remark 4.141 state that decidability of the equational theory of Da for infinite cz is an open problem. 4. The axiomatisation of IGa was established in [And01l, using the axiomatisation of IDa. The axioms are Sahlqvist equations, so IGa is canonical. Decidability of the equational theory of Ga for cz < co was proved by N6meti [N6m95, theorem 4.3]. Among other things, decidability of the universal theory of Ga for finite cz was proved by N6meti in [N6m96, theorem 4.11; remark 4.14 of this paper conjectures that the universal theory of Ga for infinite cz is also decidable. For infinite cz, it is not known whether IGa is a variety, or even closed under ultraproducts. (This was asked in [NEm96, AndGol+98, And01 l.) We also lay out some simple properties of the classes, which will be occasionally useful. PROPOSITION 5.39 Let o~ > 3 be an ordinal.

1. D a - Crsa n M o d { c i d i j - 1" i,j < o~}. 2. IGa c IDa C ICrsa. 3. Ga g CAa g ICrsa.

5.5. Relativised cylindric algebras

185

4. CAa n C r s a -- C A a N D a , andat leastfor(~ = 3, CAa n D a ~ CAa n G a . 5. GaNMod(C4) C DmNMod(C4) C_ CrsmnMod(C4, cidij : 1) C CAm, where C4 denotes the CAa axiom CiCjX - r for all i, j <

Proof (sketch). 1. It is clear that Da ~ cidij 1 for all i, j < c~. Conversely, if A E Crsa satisfies these equations, then for any x in the unit 1 of A and i, j < ~, x E r so there is y ---i x with y E dij. Thus, y c 1 and Yi : yj, so x o [i/j] -ycl. -

2. Clearly, Ga c_ Dm C Crsa, so the same holds for the closures of these classes under isomorphism. That Ga =/=Da can be shown by constructing an algebra in Da as follows. We take o~ - 3, let x - (0, 1,2) c 33, and let X be the smallest subset of 33 containing x and closed under substitutions [i/j] for i, j < 3. Let A have domain ~o(X). It can be checked that A E D3. But A does not satisfy the axioms of (4), because cocl (S~ s~co{x} 0 ~ {x}. So A r IG3. 9

2

0

1

-

To see that Dm -7/=Crsm, choose any A c Crs~ whose unit is not closed under substitutions. By the argument of (1), A does not satisfy the axioms r 1, so certainly A ~ IDa. 3. Consider the algebra A c G2 with domain go(2 {0, 1})U go(z{ 1,2})U go(2{2, 3}). Then since (0,1)--o (2,1) - l (2,3), we have A ~ {(0, 1)} _< cocl {(2,3)}. But (0, 1) -1 (x,y) =o (2,3)implies (x,y) = (0,3) ~ 1. So A ~: {(0, 1)} < clco{(2,3)}, A ~ Vx(coclx : clcox), and A ~ CA2. A similar argument works for larger c~. We are assuming c~ > 3, and in this case, CA~ g ICrsa" see [HenMon+85, 5.5.9]. Note however that CA2 c ICrs2 [HenMon+85, 3.2.61, 5.5.6]. 4. By [HenMon+71, 1.3.2], CAm ~ cidij 1. So any algebra in CAa n Crsa satisfies this axiom and hence is in Dm by (1). We will see that CA3 n D3 b CA3 N G3 in fact 5.46 below. --

5. The first inclusion is trivial, the second follows from (1), and the third is simply a matter of checking the CAa axioms, cidij 1 is needed to prove the djk <_ ci(dji "dik) part of C6. --

There are finite-schema analogues of the finite axiomatisation results, for infinite c~. For further information on these classes (there is much), see [HenMon+81, HenMon+85, N6m85, N6m86, AndTho88, N6m91, ResTho91, N6m95, Mar95, N6m96, MarVen97, And97a, AndGol+98, Mon00, And01 ], for example.

Chapter 5. Relativisation and cylindric algebras

186

Exercises 1. Which parts of fact 5.20 are valid in Da or Ga? 2. Is the analogue of proposition 5.37 true for Crs, G? 3. Let L be a finite signature and let K be a class of L-structures with K = SK and with decidable universal theory. Show that there is an algorithm to decide whether a finite L-structure is in K. [Cf. corollary 18.27.] Deduce that for n < co, it is decidable whether a finite CAn-type algebra is in ICrsn, IDn, or IGn [N6m96, theorem 4. l(ii, v)].

5.6

Relation algebra reducts of cylindric algebras

Cylindric algebras are an attempt to algebraise relations of arbitrary arity, just as relation algebras are for binary relations. We now examine the connections between them; this will be followed up at greater length in chapter 13.

5.6.1

Neat reducts and relation algebra reducts

There is a well known method of obtaining a relation-type algebra 9~a(C) from an c~-dimensional cylindric algebra C (for any ~ _> 3): 9~a(C) is constructed by taking the two-dimensional elements of C and using the spare dimensions to define conversion and composition. 9~a(C) is called the relation algebra reduct of C. More formally, this is done as follows. D E F I N I T I O N 5.40 [HenMon+85, 5.3.7] Let C be any m-dimensional cylindric algebra, where c~ >_ 3. 1. For 13 < ~, the neat ~-reduct of C (in symbols, 9"tt~C) is the l-dimensional cylindric algebra with domain {a c C" cja - a for all 13 < i < ~} and with operations + , - , 0 , l,cj,djk for j , k < [5 induced from C. 2. The relation algebra reduct of C m in symbols, 9~a(C) (dom(92r2C), O, 1, + , - , 1 , , ), ~

o

where 9

+,-,0,1

are as i n C

9 1' -- d01 (E

cYSt2C)

9 conversion is defined by ~ - ~2~0~1 ~0~1~2r, for r c 9"tt2 C

is the algebra

5.6. Relation algebra reducts o f cylindric algebras

187

9 composition is defined by r;s - c2(slr 9sOs), for r,s E ~ t 2 C . We generally identify notationally the algebras 92t1~C, ~ a C with their domains. L E M M A 5.41 92t~ C and ~ a ( C) are closed under these operations. Proof First we consider 92rBC. Clearly, 0, 1 E 92tBC. In cylindric algebras we have ckdij -- dij whenever k ~ i, j. Hence dij G r C for all i, j < I]. Let r, s E 92r[~C, and ~ _< i < ct. Then using the axioms defining CAn (definition 5.16), we have ci(r + s) - cir + cis - r + s, and c i ( - r ) - - c i r - - r (see fact 5.17 or [HenMon+71, 1.2.1 1]). So 92t1~C is closed under the boolean operations. Also, for j < [~ < i, we have c i c j r - c j c i r - cjr, using commutativity of ci, cj in CA~ (axiom C4): so 9"h:BC is closed under appropriate cylindrifications, too. Now consider N a C . By the above, it is closed under the boolean operations and contains 0, 1, and 1' - d01. Let r, s E 9~ctC and i :> 2. If i _> 3, then because ci commutes with all three substitutions (fact 5.20(6)), c i r - - CiSoS 1 2 Os2 rl __ SoSl20szci r l __ s 200s1l s z r - ~. If i-- 2, then c z s ~ x - st~ for any x E C (fact 5.20(8)), so again, c 2 ~ - ~. Finally, consider ci(r;s) -- c / c 2 ( 5 1 r 9sOs). I f / - - 2, then clearly this is c2(s~r. sOs) - r;s by idempotence of cylindrification (an easy consequence of axioms C2, C3; see fact 5.17). If i > 3, then ci commutes with c2 (axiom C4), so the above is C 2 C / ( s l r 9sOs) - - C 2 C i ( S 1 C i r . sOcis). As Ci commutes with the two substitutions here, this is equal to c2ci(cis~r, c/s~ By axiom C3, this is c2(cis~r, c/s~ Wrapping up again, this is c 2 ( s l c i r 95~ - r;s, as required, t3

The following result will be needed (and generalised) in chapter 13. Recall that bool(C) denotes the boolean reduct of the cylindric algebra C. L E M M A 5.42 I f m < n < o and C E CA,,, then bool(92tmC) C_c bool(C). Hence, if C is atomic then so are 92tmC and (if n >_ 3) 91aC.

Let S C_ 92tmC and suppose ]~9z~,,c S - G exists. Assume for contradiction that d E C and s < d < G for all s E S. Let

Proof

def 'l~ - - G - - - C m C m + I . . . C n _

We claim that x E r

l(-d).

and s < x < (5 for all s E S; this will contradict G =

Eg~rmCS. We use (5 E r have: Ci~

C throughout. First, we show that x E 92tm C. If m < i < n, we

--

Ci(CiG'--Cm...

--

CiO" Ci--Cm . . . Cn-1-d

Cn_ l - d )

:

ci(~" ci-CiCm

:

CiG" --CiCm...

... Cn-1 Cn-1 -d

=

G. -Cm... C n - l - d

----

~.

-d

as G -

CiG

by CAn-axiom C 3 by fact 5.17(1), axiom C4 by fact 5.17(5) as before

Chapter 5. Relativisation and cylindric algebras

188

This holds for all i, so a; E r C. Second, if s C S, we show s < t:. We know s < or. Also, s < d, so s . - d - 0. Hence 0 -- Cm . . . Cn-1 (s . - d ) - S " Cm . . . C n - l - d . We obtain s < - C m . . . c , , - 1 - d . So s < x as required. Finally we check that I; < ~. If not, then we have 1: - ~, so ~ < -Cm 999 cn- l - d , SO ( I . r --0. Moving the cylindrifications over, using fact 5.17(2), shows that c r - - d - 0. This contradicts d < ~. The last part of the lemma follows immediately from lemma 2.16. []

5.6.2

Relation algebra reducts and canonical extensions

The following result is due to Henkin and Monk (cf. [HenMon+71, 2.7.23] for cylindric algebras) and we will need it in chapter 13. T H E O R E M 5.43 Let o~ >_ 3, a n d let A be a non-associative algebra. I f A c 9~ct B f o r s o m e B c CA~, then A + c C H a ( B +) up to isomorphism. Note that since CAa is canonical, B + c CAa, so that ffta(B +) is defined. For C c CAa let R(C) be the expansion of C by a constant 1' and function symbols" ;defined by 1 ' - d01 , ? - - S0S 2 01S2I C, and c ' d - c2(s~c 9 sOd), for c , d E C. By facts 5.17 and 5.20, R(C) is a BAO. We claim that R(C) + - R(C § Since R(C) is just an expansion of C, R(C) ~and R(C +) have the same reduct to the signature of C. Now

Proof

(x;y.-r176

+ (r

s~

9- ( x ' y ) ) - 0

is a Sahlqvist equation and is valid in R(C), so by theorem 2.95, it is valid in R(C) § too. So for any a , b E R ( C ) § a "R(cI§ b - c c+ (s~ c+ a 9s ~247b). This says that the identity map from R ( C ) § to R(C +) preserves ';'. It can be shown similarly that it preserves 1' and ". Hence, R(C) § - R(C § as claimed. Let R ( C ) ILRA be the reduct of R(C) to the signature of relation algebras. Then obviously (R( C) Ft~ )+ - (R( C) + ) [LR A -- R( C + ) [LR a . Now A is a subalgebra of R(B) IL~. By the above and theorem 2.71, the map t ~ 9A t ~ R(B+)rt~RA given by t § (S) - {13 E U f ( B ) ' I 3 A A E S} is a complete embedding. To check that it embeds A § into ffta(B +), it only remains to check that if 2 < i < ~ and S c A +, then in R ( B +) we have ci(t + (S)) - t + (S). For '_~', we recall (section 5.3) that C A a is canonical; hence, B + E CAa, and B + ~ r >_ x. We prove the converse inclusion. Given 13 E ci(t § (S)), there must be some "/E Uf(B) with '/A A c S and 13 _< r B + ~" By fact 5.17, ~ < CiB + [~. S o { c i b " b c ~} c_ "[. So if b c 13N A then b - cib c ~[, whence 13N A - "/N A c S and 13 c t + (S). So t + 9A + ~ ffta(B +) is a complete embedding. Hence, up to isomorphism, A + c_ c f f t a ( B + ) .

[]

5.6. Relation algebra reducts o f cylindric algebras

5.6.3

189

Relation algebra reducts are relation algebras

Clearly, if C c CAm and [3 _< ct then 9~r[3C E CAIn. We will now prove a similar result for relation algebra reducts, using the work on substitutions of the preceding section.

THEOREM 5.44 (Henkin-Tarski, [HenMon+85, 5.3.8]) For any ordinal ct > 4, if C E CAm then 91a(C) is a relation algebra.

Let t~ > 4 and C E CAm. We check that 9~aC satisfies the relation algebra axioms of definition 3.8. Let x, y, z E r 2 C. We will frequently use that x -- c2c3x, and similarly for y, 2, .9, x;y. We use the following standard notation: Proof

ks(i, j)def --

k ij SiSjSk,

for distinct i , j , k < tx.

By definition, we have 2 - 2s(0, 1)x. R0. Clearly, bool(91aC) _c bool(C), so the boolean axioms are satisfied in 91aC. R2. ( x + y ) ' z - -

c2(s{(x+y).s20z) -- c2(s21x, s2~

R3. x; 1' -- c 2 ( s l x 9 s~

s~

--x;z+y;z.

) - c 2 ( s l x 9 d12) - s21s2xl _ Sl2X - x.

R4. Let w be the s-c-word 2s(0, 1)c2c3. Note that 2 - wx E 91aC. Calculation shows that ~ - c ~ - Ida\{2,3} and that ( i , j ) c (ww)* iff i - j iff ( i , j ) c (c2c3)*. Thus, ww ~,, c2c3, and ]ct \ rng(~'w)l > 2. By theorem 5.25, ww C2C 3, SO X -

WWX-

C 2 C 3 X - X.

R5. Using fact 5.20(2), we have (x + y)" -- 2s(0, 1)(x + y) -- 2s(0, l)x + 2s(0, 1)y -- X + 37. R6. First, note that s l 2 s ( 0 , 1)r162 ",-' 35(0, 1)S0C2C3, and if v is either of these words, let \ rng(v-')l > 2. So by theorem 5.25, we have s~2s(0, 1)c2c3 3s(0, 1)s0c2r . In the same way, we obtain s~ 1)c2c3 ~ 3s(0, 1)slc2c3 and 2s(0, 1)c2c3 ~ 3s(0, 1)c2c3. Hence, --

c2(s{(2s(0, l)y).s~

=

C2(3S(0, 1)s0y 9 3S(0, 1)s~x)

1)x))

by definition as s~2s(0 , 1)c2c3 "" 3s(O, 1)sl-iCzC3

= -

c23s(0, 1)(s~x 9s2Y 0 ) 3s(0, l)c2(slx .sz~ 3s(0, l)(x;y)

by fact 5.20(2) by fact 5.20(6) by definition

=

2S(0, l ) ( x ' y )

as 3s(0, 1)c2c3 "" 2s(0, 1)c2c3

=

(x'y) ~

by definition.

190

Chapter 5. Relativisation and cylindric algebras

R7. We prove that f f ; ( - ( x ; y ) ) . y - 0. By theorem 5.25, we have s~2s(0, 1)C2C 3 3s(0,2)s~c2c3, and 3s(0, 2)s~ ~ c2c3. So slff.y

-

-

= = _< =

sl2s(O, 1)x'y 3s(O,2)s~x. 3s(O, 2)s~ 3s(O,Z)(s~x.s~ c3s~ 9sOy) s~

by definition by the above by fact 5.20(2) by definition of s3,s 2 by fact 5.20(6) and c3(x;y) - x;y.

So by the cylindric algebra axioms C 1, C3, and c2y - y, we obtain the required .~;(-(x;y))-y -- c2 (sly 9s20-(x ;y)).y = c2(sl~ 9-s20(x;y) 9y) -- c20 -- O. R1. Finally, we prove associativity. As with conversion, we redefine composition equivalently using the third dimension. It is easily checked using 21 ~" s~c2c3 and s2sOc2c3 fact 5.20 or theorem 5.25 that c3s~c3 ~-"c3c2, s3s2c2c3 $0r 3. So by fact 5.20(2,6), x ; y - c3(x;y) = c3c2($21x- s ~ - C3S2C3(slx 9 sOy) _ C3(2S3S2 x.l S2S0y)- C3(S~X"sOy). (Cf. lemma 13.31 later.) Now, we get x;(y; z) - (x ;y) ; z, because

x;(y;z)

= c 2 ( s ~ x " s0c3(s~y 9 S~Z)) = c 2 ( c 3 s l x 9 c 3 s 0 ( s ~ y 9 S0Z)) -- c 2 c 3 ( c 3 s l x 0 1 0 1 0 I 0 s 2 (s3Y" s 3 Z) ) : c 2 c 3 ( s2x" s 2 s3Y" s 3 Z), and

9

(x; y ) ; z - c3 (s~c2 ( s i x . sOy) . sOz) _ c3 (c2s~ ( s i x . s2Y ) 0 . c 2 s O z ) _ c3c2 (s 13( s ~ x . sOy). c2sOz) _ c 3 c 2 ( s l x . s3s2y . 1 0 sOz), and by fact 5.20(10), So2 s3Y ] -- S~ sOy. []

5.6.4

T h e c l a s s e s S92t13CAa and Sg~ctCAn

DEFINITION 5.45 Let o~ be an ordinal and let K c CAa. 1. For 13 < or, 92tpK denotes the class {92tpC 9C E K}, and STtrpK the closure of 92t13K under subalgebras, as usual. In particular, S92t13CAa - { B" B c_ 92t13C for some C c CAa }, the class of subalgebras of neat ~-reducts of ~dimensional cylindric algebras. 2. For (x > 3, we define 9~aK to be the class {f.aaC" C c K}. The class Sg~ctK is the closure of 9~aK under subalgebras. In particular, sg~aCAa - {A" A c 9~aC for some C c CAa}, the class of subalgebras of relation algebra reducts of (x-dimensional cylindric algebras.

5.6. Relation algebra reducts of cylindric algebras

191

Some authors use the notation 9~a*CAa instead, to denote that the range of the (proper class) map 91ct is intended (and similarly for neat reducts). We see no ambiguity in 91aK when, as will usually be the case, K is a proper class and so not an algebra. The classes S~aCAn for finite n > 3, and to a lesser extent the corresponding neat reduct classes, are a major object of study in chapters 13 and 15. For now, we list some results about them. FACT 5.46

1. By theorem 5.44, Sg~aCA4 _c RA; in fact, RA - sg~ctCA4. See [HenMon + 85, 5.3.17] for a stronger result. 2. One might guess that SglaCA3 - SA, or perhaps WA. But the relation algebra axiom R4 (x -- x), among others, fails in Sg~aCA3. Hence, SgqaCA3 NA, so by exercise 1 below, SgqaCA3 3 SgqaCA4. See [Sire97] for more information about SgqaCA3; we quote some of the results in theorem 6.7 below. It has long been known that the condition t~ ___4 in theorem 5.44 is necessary for 9qaC to have associative composition. But R1 is not the only relation algebra axiom that fails in Sg~aCA3. For example, in [Sire97, theorem 4.10], an example is given of a 3-dimensional cylindric algebra satisfying the merry-go-round axioms (see fact 5.38) but whose relation algebra reduct does not satisfy R7. (This example further shows that R6 F/R7.) Monk and Fuhrken proved [Mon61b, theorem 9.10] that R A - SgqaCA3 n Mod{Rl, R4, R6}. N6meti and Simon [N6mSim97, Sim97] improved this by showing that a subset of the RA-axioms defines RA within SgqaCA3 (or within 9qctCA3) iff it includes {R1, R6} or {R1, R7}, and that a subset of the SA axioms defines SA within SgqaCA3 (or within 9qaCA3) iff it contains R6 and R7. See also theorems 6.7 and 6.8 later. We use these results in theorem 13.49 to show that SA - Sgqa(CA3 N G3). Since it is easily seen that if C c G3 then MaC ~ R7, this establishes that at least for c t - 3, CAa N Da ~ CAa N Ga. Cf. proposition 5.39. 3. If 1 < t~ < 13 then 91taCAi3 is not closed under forming subalgebras. This was proved by N6meti in [N6m83], solving problem 2.11 of [HenMon+71 ]. Furthermore, this class is not even closed under elementary subalgebras, and hence is not an elementary class [Say01, Theorem 1]. [SayN6m01] shows that its elementary closure is strictly contained in S91taCAI3. This paper also extends N6meti's result to Pinter's substitution algebras and quasi-polyadic algebras (see chapter 6 for some details of these), and shows that in contrast, the neat reducts of infinite-dimensional polyadic algebras form a variety.

Chapter 5. Relativisation and cylindric algebras

192

4. Analogous results are known for relation algebra reducts. Maddux and N6meti independently proved that 9~ctCAn C sg~aCAn for n > 4 [Madd90a, N6m86]; Simon proved the same for n = 3 in [Sim97]. 5. For ordinals c~ > 13 _> 3, sg~aCAa c_ sg~aCAI3. For finite n > 4, S9~aCAn+l is not finitely axiomatisable over Sg~aCAn. See exercise 1 below, and chapter 15. 6. 1"-]3_<,, < coS ~ a C A n - R R A (see proposition 13.48 and [HenMon+85, 5.3.13, 5.3.16]). It follows by exercises 1-2 below that for any infinite ~, we have Sg~ctCAa - RRA. (Cf. [Mon6 l b, theorem 4.1] for a corresponding result, due to Henkin and Tarski, for cylindric algebras.) For this reason, we are mainly interested in sg~aCA,, for finite n. 7. Analogous results for neat reducts hold, more or less. For any m < co, CAm -- Sr ~ SCJql:mCAm+l ~ 5921:mCAm+2 ~ . . . . Andr6ka showed that 892tmCAm+ 1 is finitely axiomatisable [And90a], but (see chapter 15) for finite n > m, S92rmCA,,+1 is not finitely axiomatisable over S92cmCA,,. We will see that the classes SO21:mCAn are associated with n-variable proof theory and that these non-finite axiomatisability results have consequences in that field. By the neat embedding theorem of Henkin-Tarski (see, e.g., [Mon61b, theorem 4.1]), S92r~CA[~+a- ~

S92r13CAI3+, , - RCAI3

n,<0)

for any infinite or. P R O B L E M 5.47 (N6meti--Sayed Ahmed) Is 9~ctCAa elementary for ot >_ 3? We now show that for all ordinals ~ > 3, Sg~aCAa is a canonical variety. The analogue of this for S92tf~CAa was proved by Monk [Mon61a]; a proof is also given in [HenMon+71, 2.6.32(ii)] and we modified it for the proof here. P R O P O S I T I O N 5.48 For ot > 3, SfftctCAa is a canonical variety.

Proof

We show that H S P ~ a C A a c_ SfftaCAa. Evidently, if ~ C CAa, i C I, then 1-Iicl ~ a ~ --- ~clI-Iicl ~ c 9~aCA~. From this we see that Pg~aCA~ c_ 9~aCA~, and hence, sPg~aCAa c_ Sg~ctCAa. So it suffices to check that Sg~aCAa is closed under homomorphic images. By the results of section 2.5.2, we can work with ideals instead of homomorphisms. Let A c_ N a B for B E CAa, and let I be an ideal of A (definition 2.36). Plainly, I is a subset of B. Let J be the ideal of B generated by I (i.e., the intersection of all ideals of B containing I). By [HenMon+71,

5.6. Relation algebra reducts of cylindric algebras

193

theorem 2.3.8], J - {b E B" b < Cio...Cil(XO-~-'''-~Xk_I) for some i0,...,it < ot and x0,...,xk-1 C I}. Now since I is an ideal of A, it is closed under +, and if x C I then 1 ;x C I and x; 1 c I. By fact 5.20(2, 7, 5) and the fact that c2x - x,

: c2sOc2x-- CoS2C2X-- COC2X-- COx.

So cox c I, and similarly, clx c I. So the above expression simplifies to J - {b E B" b < x for some x E I}. It follows that J f3 A - I. Now define a homomorphism from A / I into ffta(B/J) by a/l H a/J (for a C A ) . As J O A - I, this map is one-one. Since B/J is a homomorphic image of B, we have B/J E CAa and A / I c S ~ a C A a . So Sg~aCAa is closed under H, S, and P. By Birkhoff's theorem (2.45), this shows that SfftaCAa is a variety and can be equationally axiomatised. If A E SfftaCAa, let C E CAa with A c 9~aC. By theorem 5.43, A + c_ fftaC +. As CAa is a Sahlqvist variety, it is canonical, so C + E CAa too. Hence A + c S~ctCA~ and sg~ctCA~ is canonical. []

Exercises I. Show that if 3 _~ [~ < ~ then SgqaCA~ c Sg~aCA~. 2. Show that RRA - Sg~aRCA~ c SgqaCA~ for all ordinals o~ > 3. 3. For a class K of BAOs, let S d K denote the closure under isomorphism of the class of dense subalgebras of algebras in K. Show that (a) sg~aCAo~ - ["]3<,,_ 3, where C,, E CA,,, embed A in an ultraproduct of arbitrary expansions of the C,, to the signature of CAo~. For (3b), consider the subalgebra of the ultraproduct generated by the image of A under this embedding.] 4. Check the congruences 25(0, 1)r162 ~' r162 etc. in the proof of theorem 5.44. 5. Prove theorem 5.44 using only fact 5.20. (See [HenMon+85, 5.3.8-9] for a solution.) 6. For finite n, show that if C c CA,, is atomic then the set of atoms of r m C is {CmCm+l . . . C n - l X ' X an atom of C}. (By lemma 5.42, r is atomic.) 7. Show that for 13 < c~, STttl3CAa is a variety.

194

Chapter 5. Relativisation and cylindric algebras

5.7

Relation algebra reduets of other cylindric-type algebras

For D c Da and 13< c~, we might try to define 92tl3D to be a CAi3-type algebra with domain {x c D" cix - x for 13 < i < ~x}, and operations induced from D as before. However, this may not be closed under the ci for i < ~. Similarly, the problem we meet in trying to define 9~aC for D E Da or D c Ga is that {x C D" c i x - x for i > 2} may not be closed under composition as defined by x;y -- c2(s21x9s2Y o ). There are alternative definitions known (see the exercises), but they do not necessarily yield associative composition. So we abuse notation somewhat, by the following definition. D E F I N I T I O N 5.49 Let ~ > 3.

1. Let A be a relation-type algebra, and let D c Da. A relation algebra reduct embedding from A into D is a boolean algebra embedding t 9 bool(A) bool(D) satisfying, for all r,s E A: (a) cit(r) - t(r), for all i with 2 _< i < ~. (b) t ( l ' ) - d0 .

(c) t(P) -- s2s~ (d) t ( r ' s ) - c2(slt(r).s~ 2. We write Sg~aDa for the class of all relation-type algebras A such that there is a relation algebra reduct embedding from A into some D c Da. 3. We define sg~aGa similarly. R E M A R K 5.50

1. Of the many notation abuses in the book, sg~aDa and Sg~aGa are probably the worst. They behave similarly to Sg~aCAa and are nice to use, once used to them; and they are very important classes. But the reader should always be aware that they do not denote the result of successively applying 9~a and S to algebras in D~,G~, and indeed that for C in D~ or G~, 9~aC is in general undefined. To limit confusion, we will not go on to write S ~ a K for subclasses K of Da, Ga m except of course when K c_ CAa, in which case definition 5.45 applies. 2. Of course, sg~aDa and sg~aGa are closed under taking subalgebras. So although we have not defined the classes 9~aDa, 9~aDa, in a way the definition behaves as though we had.

5. 7. Relation algebra reducts o f other cylindric-type algebras

195

3. Unlike for cylindric algebras, we have not been able to define 9:taC for an algebra C c Da (or Ga). (The nearest we might get is to define the class Sg~a{ C}.) Because of this, we do not find definition 5.49 entirely satisfactory, and we set a problem (5.55 below) to improve it. 4. Suppose that we were to extend definition 5.49 in the natural way to subclasses K of Da, Ga. Suppose we have such a K, and K c_ CAa too. If A c Sg~aK, then there would be some D c K and a relation algebra reduct embedding t" A ~ D. Then as D E CAa, 9~aD is defined according to definition 5.40, rng(t) c_ 9~aD, and (if a >_ 4) t" A -~ 9~aD is a relation algebra embedding; so A c S ~ a K by definition 5.45. Conversely, if A c sg~aK as in definition 5.45, then there is C c K with A c_ 9~aC. Clearly, the inclusion map from A to C is a relation algebra reduct embedding, so that A c Sg~aK via the extended definition 5.49. The conclusion is that there is no potential clash of notation between definition 5.45 and the natural extension of definition 5.49 to subclasses of Da ACAa. T H E O R E M 5.51 Let ~ >__3, and let A be a non-associative algebra. If t " A ~ B is a relation algebra reduct embedding, f o r some B c Da, then t + 9A + --~ B + (as defined in theorem 2.71) is a complete relation algebra reduct embedding. Proof

The proof is similar to that of theorem 5.43" we leave it as an exercise.

[]

We will see in chapter 13 that for finite ~ >_ 4, these classes S ~ a D a and Sg~aGa are well behaved and indeed well known by another name: the canonical varieties RAa, as defined in [Madd83]. In particular, Sg~aD4 - Sg~aG4 - RA. Also, Sg~aG3 - WA. (We will show in theorem 13.49 that S9%t(CA3 N G3) - SA.) Here we begin by proving the analogue of theorem 5.44. NOTATION 5.52 For an c~-tuple s o f elements o f some set, and i, j < ~ we write xi f o r the ith entry o f 2, and s f o r the o~-tuple (x0,... ,Xi-l,Xj,Xi+l,... ). For example, "=201 _((220)0)1 _ (Xl,XO,XO,X3, . . . . ) ~012 L E M M A 5.53 Let ~ >__4. Then S ~ a D a c RA. Proof Let D E Da. We let D be a cylindric relativised set algebra of a-ary relations on the set M. Let A be a relation-type algebra that embeds in D by a relation algebra reduct embedding. We may assume without loss of generality that dom(A) c_ dom(D). We check that A E RA by running through the relation algebra axioms. Let a,b, c E A , and let .g be an arbitrary element of the unit 1 of D. Note that for i, j < cx, s E s~a iff ~j c a (because ~ c 1); see notation 5.52.

196

Chapter 5. Relativisation and cylindric algebras

R0. As b o o l ( A ) C_ b o o l ( D ) , b o o l ( A ) is a boolean algebra. R2. Y E (a + b ) ; c - c2(s~(a + b ) . s~ iff 37 E s l ( a + b ) - s ~ for some 37 E l, 37 =2 ~, iff 37~ E a + b and ~2 E c for s o m e 37 - 2 ~ with 37 E 1, iff (37~ E a or 37~ E b) and ~2 E c, for s o m e 37 =2 x with 37 E 1, iff .~ E a ; c + b;c. Hence,

(a+b);c-a;c+b;c. R3. Y E a" 1' -- c2(s21a 9s~ ) iff there is 37 --2 ~ with 37 E 1,3721 E a, and ( ~ ) 0 -(~2)1. Such a 37 must satisfy Y2 - Xl - - i.e., we must have 37 - ~ . Since indeed ~2 E 1, we see that Y E a ; l ' iff 37~ -- (~2)~ _ ~ E a, iff Y E a as a -- c2a. So a ; 1' -- a. R4.

y Ea

2 0 1 2 0 1

~201201 s2a iff.~012012 E a. Evaluating, this is iff y2 E a, iff g E a as a is 2-dimensional. We conclude that ~ - a. -- $0S1S2SoSl

R5. Y E (a + b)" -- ~0~176 ~2~o~1 (a + b) iff =2Ol - 01 =201 E b, iff a012 E a + b, iff-~o12 E a or -r

:zE~+b. R6. Using that c~ >_ 4 and (a'b) ~ is 2-dimensional, we have .~ E (a'b) ~ iff Yo3 E -y3201 ---237, Y2 -1 E a, (a" b)~ _ sosl20s2c21(sla - sOb), iff for s o m e 37 E 1 we have-~0012 and ~2 E b. Since y -

(xl , x o , Y 2 , X O , X 4 , . . .

), this is iff

( , ) for some m E M we have y -

(Xl ,xo, m , x o , x 4 , . . .

) C 1,

(X! , m , m , x o , x 4 , . . . ) E a, and (m, x o , m , x o , x 4 , . . . ) E b. Similarly, as b ' 6 is 2-dimensional, Y E/9 ;6 iff 23o E/," 6, iff (**) for some n E M we have (xo,xl ,tl,XO,X4,... ) E 1,

(n,xo,xo,xo,x4,... ) E b, and (Xl , n , n , x o , x 4 , . . . ) E a. Since 1 is closed under substitutions and b is 2-dimensional, we have

(m,xo,m,xo,x4,...) E b (n,xo,Xo,Xo,X4,...)

Now it is clear that (**) r

C b

( m , x o , x o , x o , x 4 , . . . ) E b, ( n , x o , n , x o , x 4 , . . . ) E b.

( . ) , as we can let m - n.

R7. We prove that 6 " ( - ( a ' b ) ) . b - O. A s s u m e for contradiction that there is some ~ E 6 ; ( - ( a ' b ) ) . b. By 2-dimensionality, we may suppose that :co - x 3 . Then ( x o , x l , x 2 , x 3 , . . . ) E b, and for some m E M, we have y (X0,Xl , m , x 3 , .

. . ) E 1, ( m , x o , x o , x 3 ,

. . . ) E a, a n d ( m , xl ,b, x 3 , . . . ) E - ( a "

b).

So there is no n E m with (m,xl,n, x3,... ) E 1, (m,n,n,x3,. . . ) E a, (n,xl ,n, x 3 , . . . ) E b.

5. 7. Relation algebra reducts of other cylinclric-type algebras

197

N o w take n - xo - x3. By the above, we have (m, xl , n, x3, . .. ) - y 0,223C1, (m,n,n,x3,. .. ) = (m,xo,xo,x3,.. . ) c a, and (n,xl,xe,x3,.. . ) - ~ c b. As b is 2-dimensional, we obtain (n,xl , n , x 3 , . . . ) E b, a contradiction. R1. It remains to prove associativity. L e t ~ E ( a ; b ) ; c ; we show ~ c a;(b;c). By 2-dimensionality of (a ;b) ;c and a ;(b; c), we can assume that x3 - Xl. There are m,n E M with both 37 -- (xo,xl ,m,x3,... ) and E - (xo, m , n , x 3 , . . . ) in the unit 1, and (xo,n,n,x3,...) C a, (n,m,n, x3,... ) C b, (m,xl , m , x 3 , . . . ) C c. Now .~ --2 z' def (xO,xl,n,x3,...) -- E1 C 1 and (~,)1 _ (xo,n,n,x3,...) C a. But also, (E') 0 -- (n,xl ,n,x3,... ) --2 (n,xl,m,x3,. -" ) -- z213--~ C I. Plainly, ( n , m , m , x 3 , . . . ) C b a n d ( m , x l , m , x 3 , . . . ) C c, so we have (~/)0 c b;c. Thus, E sla.s~ and .~ c a;(b;c), as required. The converse is similar. [] In chapter 13 we will see that R A - Sg~aD4 -- Sg~ctG4 and SA - Sg~a(CA3 NG3). We can prove similarly that SgqctG3 c WA. Exercise 4 below shows that WA c SgqaG3, so WA - SgqaG3 in fact. LEMMA

5 . 5 4 SgqaG3 - W A .

Proof For 'C_', let ~ E G3 be a cylindric relativised set algebra of 3-ary relations on the set M, say, and let .,q be a relation-type algebra that embeds in ~ by a relation algebra reduct embedding. As in lemma 5.53, we assume that dom(.,q) C_ d o m ( G ) , and check that .,q c WA. As before, bool(A) c_ b o o l ( ~ ) so b o o l ( A ) is a boolean algebra. R 2 - R 5 are proved as in the lemma, since only dimensions 0, 1, 2 were used there. To prove R6, R7, and WL, let a,b E .,q, and let s - (x0,xl ,x2) be an arbitrary element of the unit 1 of ~. R6. We have ~ c (a ;b) ~ iff (Xl ,x0,x0) C a ;b, iff there is m E M with (xl ,x0,m) c 1, (xl,m,m) c a, and (m,xo,m) c b. Since I is closed under substitutions and permutations and b is 2-dimensional, this is iff there is rn E M such that (xo,xl,m) C 1, ( x l , m , m ) C a, and (m,xo,xo) E b. Equivalently, there is m E M with (xo,xl,m) C 1, (xo,m,m) C [~, and (m,xl,m) E 6. This is equivalent to ~ c/~;~, as required. R7. Assume for contradiction that ~ E ~ ; ( - ( a ; b ) ) . b. So there is m c M with (xo,xl,m) E 1, (xo,m,m) C d, and (m,xl,m) ~ a;b. So (m,xo,xo) E a. But also, (xo,xl ,x2) C b, b is 2-dimensional, and 1 is closed under substitutions, so (xo,xl,xo) C b. The closure of 1 under permutations yields (m,xl ,xo) C 1. But now, (m,xl,x0) C 1, (m,xo,xo) C a, and (x0,xl,x0) c b witness that (m, x l, m) c a ; b. This is a contradiction, so a ,( - (a ; b) ). b ----0 as required. WL. It remains to check weak associativity, ((a. 1') ,1)" 1 - (a. 1') ; 1. Certainly, (a. 1');1 - ( ( a - 1 ' ) ; 1)" 1' C ((a. 1'); 1)" 1. For the converse, first note that

Chapter 5. Relativisation and cylindric algebras

198

because 1 is closed under substitutions, for any b c A a n d ) 7 - (y0,yl ,y2) C 1 we have 37 c b ; l iff there is m C M with (yo,Yl,m) C 1 and (yo,m,m) E b. Now assume that s c ((a. 1') ; 1); 1. So there is m C M with (xo,xl,m) E 1 and (xo,m,m) E (a. 1');1. Hence, there is n E M with (xo,m,n) E 1 and (xo, n,n) c a . 1'. Since 1' - d 0 1 , n - x0, so (xo,xo,xo) c a . 1'. Certainly, (xo,xl,xo) E 1. Hence, s (xo,xl,x2) E (a. 1'); 1. The converse inclusion is left to exercise 4.

[]

P R O B L E M 5.55 Is there a good definition of fftaD for D c Da? Composition should be defined by x ; y - c2(s~x 9s2y 0 ), as usual, so it would be more helpful to find a 'nice' K c_ Da such that fritz D is closed under the relation algebra operations, for D c K. This may be difficult, since for finite n _> 4, the equational theories of Dn, Gn are decidable (fact 5.38), while the equational theories of SfftaDn and SfftaGn (as in definition 5.49) are not (theorem 18.28 and corollary 13.47). P R O B L E M 5.56 How are Sg~aD3 and S9~a(CA3 AD3) related to NA, WA, SA, and RA?

Exercises 1. Suppose that A E R A , C E CA,, for some n > 4, and dom(A) c_ dom(C). Show that A c 9~aC iff the inclusion map t" A ~ C is a relation algebra reduct embedding. Show further that A c_c ~ctC iff t is a complete relation algebra reduct embedding. 2. Prove theorem 5.51. 3. [Andr6ka] For finite n > 4, let ~x abbreviate s 2is 3l . . . s nI - I x. L e t D c D , , . Show that ~ x Writing ~ce(s~z. boolean algebra.

~x for all x c D.

2 0 I and z'w ~(D) for { ~ x ' x c D}, define 1' - ~d01, s -- S0SlS2Z s~ for z, w c ~(D). Show that ~(D) is closed under these and the operations, so that A - (~(D), + , - , 0 , 1,1', V, ") is a relation-type Show that A is a non-associative algebra. Is it in WA? SA? RA?

4. Assuming that every weakly associative algebra has a relativised representation (it does - - see [Madd82] or theorem 7.5), prove that WA c_ SfftaG3. [Given a relativised representation M of A E WA, consider the algebra G c G3 with unit Iq - {(xo,xl,x2) E 3M " M ~ Ai,j<3 l(xi,xj)} and domain ga(1q), and the map t ' A ~ G given by t(a) - { ( x 0 , x l , x 2 ) C IG'M a(xo,xl)}, for a C A. See also exercise 13.2(1).] 5. Show that Sffta(CA3 N G3) c SA. rem 13.49.)

(In fact we have equality: see theo-