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Introduction to approximations
Introduction to approximations
355
A substantial fraction of the remainder of the book will be concerned with finitedimensional approximations to RRA. The story will take some time to develop. Here, we give a short introduction to the three main approaches to approximations. In chapter 12, we will develop the elementary properties of 'bases', perhaps the most concrete of them. The other two - - relation algebra reducts and relativised representations will be discussed in chapter 13, where all three approaches will be proven equivalent. Hierarchy results will be proved in chapters 15 and 17, undecidability results in chapter 18, and finite model property results in chapter 19.
Approaches to approximating representability When Tarski originally set down his equations defining relation algebras (definition 3.8), it must have been hoped that they would characterise the isomorphism types of proper relation algebras - - that is, that they would axiomatise RRA. This was not to be. Lyndon exhibited the first non-representable relation algebra in [Lyn50], a paper also noteworthy for introducing a 'point-by-point' approach to building representations of relation algebras which we presented in chapter 11 in terms of the game Ga(A). Later examples of Lyndon's using projective planes (section 4.5) led to Monk's famous proof ([Mon64], to be proved in section 17.1) that RRA is not finitely axiomatisable in first-order logic. We can view these examples as demonstrating that a relation algebra can contain 'contradictions'. These are invisible in small parts (with at most four points) of a representation of the algebra, which are well controlled by the axioms governing the basic operations (associativity of composition, etc.); but they show up when we try to build, in Lyndon's way, larger fragments of a representation and can destroy the possibility of a representation existing at all. For example, the proof in chapter 4 that McKenzie's relation algebra is not representable required a 5-point fragment (see figure 4.1 ). The size of fragment needed to detect 'contradictions' in a relation algebra can be arbitrarily large, and Monk used this to prove that no one first-order sentence can control them, so yielding the non-finite axiomatisability of RRA. Monk extended his result to cylindric algebras, the higher-arity analogue of relation algebras, in [Mon69]. Monk's results were very influential. One reaction to them was to try to control the behaviour of larger fragments of a representation. We may try to characterise those relation algebras where we have no problems in building up to n points of a potential representation, for some finite n. (We say 'potential' because of course the relation algebra may turn out not to have a representation. We will return to this point below.) As n increases, the classes of such algebras approach RRA. We can ask whether the classes are finitely axiomatisable, whether they form a strict hierarchy as n increases, or whether they reach RRA at some finite n. At any rate, they should be useful approximations to RRA, and shed light on it.
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Introduction to approximations
Relational and cylindric bases Two striking ways of doing this were found. In a series of publications dating from the late 1970s, Maddux used n-dimensional relational and cylindric bases, which we will discuss in chapter 12. Recall (definition 11.2) that an atomic network describes which elements of A hold between each pair of its nodes; the coherency conditions in the definition ensure that the description respects the relation algebra operations. We may think of an atomic network as the isomorphism type of an n-point fragment of a potential complete representation of A. An n-dimensional basis for an atomic relation algebra A is a set of n-point atomic networks over A, with certain closure properties; the difference between relational and cylindric bases is in the closure properties that are required. The class RAn denotes those relation algebras that have an n-dimensional relational basis [Madd83], or embed in an algebra that does. For such a relation algebra, we can use the basis to build a (complete) representation of it up to n points in Lyndon's manner; and we can even keep on going, except that for every new point we add, we must forget some other point in order to limit the total size of the fragment to n. The relational basis always shows us how to continue, because of its closure properties. Essentially, it encodes a winning strategy for 3 in an 'n-pebble' version of the game G~ of definition 11.3, in which networks are restricted to having n nodes. If the algebra has a cylindric basis [Madd78b, Madd89b], we can do even more, in that the different ways that the basis dictates to add points are guaranteed to be compatible with each other, again so long as no more than n points are simultaneously considered. This compatibility property is a very strong 'Church-Rosser condition' obtained from an amalgamation closure property in the definition of cylindric bases. It corresponds closely to homogeneous representations, and also to commutativity of quantifiers in first-order logic: the equivalence of 3x3ycp and 3y~xcp. Cylindric bases also encode winning strategies, but in a new kind of game in which V is also allowed to demand the amalgamation of two previously-played atomic networks, provided they are 'compatible' and their amalgam has no more than n nodes. The fact that bases 'are' winning strategies in games makes them very useful to us. Their combinatorial properties are accessible via the games, and their existence can be approximated by truncating these games. Maddux introduced relational bases in [Madd83] in the context of n-variable proof theory, to characterise models in which sequents are valid if and only if they are provable in a certain sequent calculus using at most n variables. Cylindric bases are used when the proof theory has an axiom for commuting quantifiers.
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Relation algebra reducts of cylindric algebras The second way to control large fragments of a potential representation of a relation algebra is due to Henkin, Monk, and Tarski. If the relation algebra is the relation algebra reduct (definition 5.40) of an n-dimensional cylindric algebra, or is a subalgebra of such a reduct, then we can arrange that fragments of a potential representation of up to n points are controlled by the cylindric algebra axioms of definition 5.16, just as smaller fragments (size four or less) are controlled by the axioms of relation algebras. The class of relation algebras that arise in this way is of course SfftaCAn (definition 5.45). On the face of it, the definition of Sg~aCAn is more abstract than that of RAn. For example, given even a finite relation algebra, it is not immediately clear how one might determine whether it is in SfftafA,, for given n. (In fact, for n > 5 this problem is undecidable: see theorem 18.13.) Maddux made progress towards a more concrete characterisation, by proving in [Madd78b, Madd89b] that any subalgebra of a relation algebra with an n-dimensional cylindric basis is in SgqaCAn. This is a useful sufficient condition for membership of SgqaCA,,. He established it by showing that an n-dimensional cylindric basis can be viewed as the atom structure of an n-dimensional cylindric algebra. The original relation algebra can be recovered from this cylindric algebra by taking its relation algebra reduct, and possibly then taking a subalgebra of that. Unfortunately, whether the condition is also necessary - - whether every algebra in sg~aCA,, is a subalgebra of a relation algebra with an n-dimensional cylindric basis - - remains an open question. So we cannot at present characterise S9~aCA,, by n-dimensional cylindric bases. Nonetheless, we will prove in chapter 13 that SfftaCA,, can be equivalently defined in the same way as RAn, using not cylindric bases but n-dimensional hyperbases. Hyperbases generalise cylindric bases by using hypernetworks instead of atomic networks. Hypernetworks contain more information than simply the relation algebra structure on n points of a potential representation. Each sequence of up to n points in an n-dimensional hypernetwork carries a 'hyperlabel'. The idea is that the points of the hypernetwork are controlled by an element of an n-dimensional cylindric algebra, and the additional hyperlabelling structure describes this n-ary relation more fully than the merely binary relation algebra structure on the points can do. We believe that this extra information is essential to capture sg~aCA,, and cannot be eliminated, but further research is needed.
Relativised representations How can we formalise the notion of 'potential representation' of a possibly nonrepresentable relation algebra? The answer is to use the relativised representations of definition 5.1. In chapter 5, we saw a natural way of defining a relativised representation for a relation algebra, or more generally a non-associative algebra,
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A. We dropped the requirement that the representation be fully classical (i.e., the original unrelativised notion of representation), and instead allowed relativised interpretations of all the relation algebra operators. The unit (the interpretation in the representation of the top element 1) is a reflexive, symmetric relation on the base set M of the representation, and all operations, in particular negation ' - ' and composition ';', are relativised to the unit. So a pair (x, y) of elements of M stands in the relation r;s (where r, s E A) if there is z E M with (x,z) standing in the relation r and (z,y) in the relation s, so long as (x,y) is in the unit. The interpretation of ' - ' is defined similarly; the other operations are unaffected by relativisation since 1M is reflexive and symmetric. Maddux used this kind of relativised representation to characterise weakly associative algebras (section 5.1 and theorem 7.5). We now want to insist that 'on the small scale', the relativised representation does appear to be classical. We may then vary the scale up to which classical behaviour is required. If we only demand that things appear classical when considering at most three points of the base of the relativised representation, then we capture semi-associativity and hence the class SA. If we require classical behaviour of four-point fragments, we capture associativity and hence ordinary relation algebras. Indeed, these are alternative ways of defining SA and RA. Requiring classical behaviour for fragments of up to five points takes us to a strictly smaller class than RA, up to six takes us to a still smaller one, and so on. The 'limit' or intersection of these classes is RRA. Thus, we obtain a whole series of 'approximations' to, or perhaps 'n-variable analogues' of, RRA.
n-square relativised representations
In fact, there is more than one way of requiring classical behaviour of fragments of a relativised representation up to a given size. The basic requirement is that the relativisation (of negation and relational composition) is only visible when fragments of the representation above a certain size (say, n points) are considered. The n-square relativised representations of definition 5.7 are like this. Recall from definition 5.7 that a subset of a relativised representation M is a clique if all pairs of elements of it lie in the unit. We defined M to be n-square if for any clique C c_ M of size less than n, if x,y C C and (x,y) lies in the relation r ' s then there is a clique C' ~ C in M containing a point z realising this composition, so that (x,z) lies in the relation r, and (z,y) in s. An n-square representation is 'locally classical', in that if we look at it through a moveable 'window' only big enough to show n points, we will never discover using the relation algebra operations that it is relativised. n-fiat relativised representations A stronger requirement than n-squareness involves commuting quantifiers. To explain it, we will use 'clique-relativised' semantics. By definition 5.1, a relativised representation M of a non-associative algebra A is a model of a certain first-order theory RA in the signature L ( A ) of
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which each element of A is a binary relation symbol. Any n-variable formula r in the first-order language in signature L(A), written with variables x0,... ,X,_l, say, can be naturally interpreted in M by relativising quantifiers of q~ to cliques. In the clique-relativised semantics of r the range of every assignment of variables to elements of M used in the inductive evaluation of r should be a clique. Equivalently, we can syntactically relativise all quantifiers in q) to Ai,j
Unifying the three approaches So we have three approaches to approximating RRA - - by bases (RA,,), relation algebra reducts (SfftctCA,,), and relativised representations. In chapter 13, we will unify all of them. We show that for all finite n > 4: 9 sg~aD,, - Sg~aG,, - RA,, is the class of all relation algebras with an nsquare relativised representation, 9 sg~ctCA,, = S91a(CA,, n D,) = Sg~a(CA,, A G,,) is the class of subalgebras of relation algebras with an n-dimensional hyperbasis, and the class of relation algebras with an n-flat (and infinitarily n-flat, and n-smooth) relativised representation.
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Thus, in each case, the same class can be equivalently characterised in terms of bases, relation algebra reducts, and relativised representations.
The 3-dimensional case
Above, we required that n _> 4. The case n - 3 is rather special, because Sg~aG3 - WA ~ RA3 (cf. lemma 5.54) and S9~aCA3 ~ NA. The proof that a relation algebra reduct has a basis requires n _> 4. But all is not lost: we can show that RA3 - SA - Sg~a(G3 f-)CA3), this being the class of non-associative algebras with a 3-square and/or 3-flat and/or 3-smooth relativised representation.
The payoff We give some illustrations of the benefits of having this three-way characterisation of the classes RAn and Sg~aCAn.
Representation theory for RA Since RA = RA4 : Sg~aCA4, a corollary of our results is that the class RA of relation algebras is precisely the class of nonassociative algebras with 4-square (or equivalently, 4-flat or 4-smooth) relativised representations. Indeed, we will see in chapter 19 that the class of finite relation algebras is precisely the class of all finite non-associative algebras with finite 4square (or equivalently, finite 4-flat or 4-smooth) relativised representations. The representation theory developed here characterises arbitrary relation algebras, and may therefore be useful for studying relation algebras without classical representations. Bases and representability Using the representation theory developed here, we can give a direct argument (proposition 13.48) showing that the RAn (and the S91aCAn) converge to RRA. A similar argument using relational bases appears in [Madd91 b, p. 112]. Canonicity We can also prove that the varieties RAn (4 < n < to) are canonical by generalising Monk's theorem for RRA (theorem 3.36) to n-square (and n-flat) relativised representations. Axiomatising RAn and SfftaCAn It is easily seen that the varieties RAn and S~ctCAn (5 _< n < o~) are pseudo-universal classes (definition 9. l) by each of their characterisations. Being contained in RA, they are discriminator varieties. So we can use the techniques of chapter 9 to give equational axioms for them in three different ways, corresponding to our three kinds of characterisation. We do this in outline in theorem 13.55. Though the axiomatisations are infinite, and for n > 5 necessarily so, they are explicit and recursive. It may be interesting to compare the axiomatisations obtained from the three different characterisations of the classes.
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A hierarchy of approximations In part IV we will be better able to assess the merits of RAn and SfflaCAn as approximations to RRA. The following diagram and text summarises some of what we will find.
NA
RA3
RA4
uS
II
II
II
Sg~a(CA3 NG3)
S9~aG4
Sg~aG5
--.
II
Jl u
-..
S9~aG3
II WA
~f
SA
~f
3
RA
II
II
Sffta(CA3NG3)
SfftaCA4
3
RA5
3
SfftaCA5 3
...
-..
RRA
RRA
Inclusions between classes All classes shown in the diagram are canonical conjugated varieties, all except NA and WA are discriminator varieties, all inclusions are strict, and all except the ~ f s are not finitely axiomatisable (i.e., the smaller class, which may be RRA, is not finitely axiomatisable over the larger). Remark 15.13 will show that RAm SfftaCA,, for all finite m,n :> 5, so no more inclusions other than those derivable by transitivity can be added to the diagram above. Also, proposition 13.48 will show (combining work of Monk and Maddux) that
n RAn- n
3
Sg~aCA,,-RRA.
3
So by exercise 1 below, all inclusions derived by transitivity from the non-finitely axiomatisable ones shown in the diagram are also non-finitely axiomatisable. For example, RRA is not finitely axiomatisable over RAs. Lemma 5.54 and exercise 5.7(4) showed that Sg~aG3 - WA. The other equalities in the diagram will be proved in corollaries 13.47 and 13.50. It is well known that SA 3 RA. The inclusions RA4 3 RA5 3 -.. were proven strict in [Madd92]. Infinitely many of the inclusions SfftaCA4 _~ sg~aCA5 _~ ... were shown to be strict in [Madd89b], and in [HirHod ~02c] and chapter 15 it is shown that they are all strict. The non-finite axiomatisability results are proved in [HirHod00, HirHod01 a] and chapters 15 and 17. For n _> 6, RAn and Sg~aCA,, are not closed under completions (see section 17.7) and are therefore not Sahlqvist varieties (theorem 2.96). The analogous questions for n - 5 are open.
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In some ways, however, RAn is better behaved than SfftaCAn: 9 RA3 - SA, but as we saw in fact 5.46, sg~ctCA3 ~ NA. (This is why we use Sffta(CA3 A G3) in the bottom line of the diagram above. The irregularity caused is mostly only visual, since S f f t a C A , , - Sg~a(CAn A Gn) for finite n>4.) 9 Given finite n _> 5, it is decidable (in polynomial time) whether a finite relation algebra A is in RAn (corollary 12.32), but undecidable whether A E SfftaCA,, (theorem 18.13). 9 For any finite relation algebra A, we have A c RAn iff A has a finite n-square relativised representation (theorem 19.18). This 'finite model property' fails for SfftaCAn and n-flat relativised representations (proposition 19.19), although by theorem 19.20 it does hold for the class Sg~a{ C c CA,, : C finite}. 9 [Ste00, SteVen98] characterised RAn as the class of subalgebras of relation algebra reducts of 'Q,,-algebras', where Qn is an nZ-ary operator symbol similar to those defined by J6nsson in [J6n91], and those in section 8.2. The Qn-algebras form a finitely axiomatisable Sahlqvist variety (see exercise 12.4(12) below). We do not know of a similar subreduct characterisation of sg~aCA,,.
Approximations or analogues? The conclusion is that RAn and S~aCA,, do not approximate RRA very closely: for any finite n >_ 5, RRA and even RA,,+I are 'infinitely far away' from RAn in terms of axioms required to make up the difference, and similarly for Sg~aCA,,. Even Sg~nCAn is infinitely far from RAn. Perhaps these classes are better seen as finite-dimensional analogues of RRA. (The 'dimension' here ('n') is connected to the number of variables used in proofs in first-order logic; we will draw out this connection later.) They are non-finitely axiomatisable canonical varieties with many of the properties of RRA, and they have a workable representation theory.
Exercises 1. Let K,, (n _< co) be elementary classes of relation algebras (say), with Ko _~ Kl _~ ... and Ko~ : f~n
355
A substantial fraction of the remainder of the book will be concerned with finitedimensional approximations to RRA. The story will take some time to develop. Here, we give a short introduction to the three main approaches to approximations. In chapter 12, we will develop the elementary properties of 'bases', perhaps the most concrete of them. The other two - - relation algebra reducts and relativised representations will be discussed in chapter 13, where all three approaches will be proven equivalent. Hierarchy results will be proved in chapters 15 and 17, undecidability results in chapter 18, and finite model property results in chapter 19.
Approaches to approximating representability When Tarski originally set down his equations defining relation algebras (definition 3.8), it must have been hoped that they would characterise the isomorphism types of proper relation algebras - - that is, that they would axiomatise RRA. This was not to be. Lyndon exhibited the first non-representable relation algebra in [Lyn50], a paper also noteworthy for introducing a 'point-by-point' approach to building representations of relation algebras which we presented in chapter 11 in terms of the game Ga(A). Later examples of Lyndon's using projective planes (section 4.5) led to Monk's famous proof ([Mon64], to be proved in section 17.1) that RRA is not finitely axiomatisable in first-order logic. We can view these examples as demonstrating that a relation algebra can contain 'contradictions'. These are invisible in small parts (with at most four points) of a representation of the algebra, which are well controlled by the axioms governing the basic operations (associativity of composition, etc.); but they show up when we try to build, in Lyndon's way, larger fragments of a representation and can destroy the possibility of a representation existing at all. For example, the proof in chapter 4 that McKenzie's relation algebra is not representable required a 5-point fragment (see figure 4.1 ). The size of fragment needed to detect 'contradictions' in a relation algebra can be arbitrarily large, and Monk used this to prove that no one first-order sentence can control them, so yielding the non-finite axiomatisability of RRA. Monk extended his result to cylindric algebras, the higher-arity analogue of relation algebras, in [Mon69]. Monk's results were very influential. One reaction to them was to try to control the behaviour of larger fragments of a representation. We may try to characterise those relation algebras where we have no problems in building up to n points of a potential representation, for some finite n. (We say 'potential' because of course the relation algebra may turn out not to have a representation. We will return to this point below.) As n increases, the classes of such algebras approach RRA. We can ask whether the classes are finitely axiomatisable, whether they form a strict hierarchy as n increases, or whether they reach RRA at some finite n. At any rate, they should be useful approximations to RRA, and shed light on it.
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Introduction to approximations
Relational and cylindric bases Two striking ways of doing this were found. In a series of publications dating from the late 1970s, Maddux used n-dimensional relational and cylindric bases, which we will discuss in chapter 12. Recall (definition 11.2) that an atomic network describes which elements of A hold between each pair of its nodes; the coherency conditions in the definition ensure that the description respects the relation algebra operations. We may think of an atomic network as the isomorphism type of an n-point fragment of a potential complete representation of A. An n-dimensional basis for an atomic relation algebra A is a set of n-point atomic networks over A, with certain closure properties; the difference between relational and cylindric bases is in the closure properties that are required. The class RAn denotes those relation algebras that have an n-dimensional relational basis [Madd83], or embed in an algebra that does. For such a relation algebra, we can use the basis to build a (complete) representation of it up to n points in Lyndon's manner; and we can even keep on going, except that for every new point we add, we must forget some other point in order to limit the total size of the fragment to n. The relational basis always shows us how to continue, because of its closure properties. Essentially, it encodes a winning strategy for 3 in an 'n-pebble' version of the game G~ of definition 11.3, in which networks are restricted to having n nodes. If the algebra has a cylindric basis [Madd78b, Madd89b], we can do even more, in that the different ways that the basis dictates to add points are guaranteed to be compatible with each other, again so long as no more than n points are simultaneously considered. This compatibility property is a very strong 'Church-Rosser condition' obtained from an amalgamation closure property in the definition of cylindric bases. It corresponds closely to homogeneous representations, and also to commutativity of quantifiers in first-order logic: the equivalence of 3x3ycp and 3y~xcp. Cylindric bases also encode winning strategies, but in a new kind of game in which V is also allowed to demand the amalgamation of two previously-played atomic networks, provided they are 'compatible' and their amalgam has no more than n nodes. The fact that bases 'are' winning strategies in games makes them very useful to us. Their combinatorial properties are accessible via the games, and their existence can be approximated by truncating these games. Maddux introduced relational bases in [Madd83] in the context of n-variable proof theory, to characterise models in which sequents are valid if and only if they are provable in a certain sequent calculus using at most n variables. Cylindric bases are used when the proof theory has an axiom for commuting quantifiers.
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Relation algebra reducts of cylindric algebras The second way to control large fragments of a potential representation of a relation algebra is due to Henkin, Monk, and Tarski. If the relation algebra is the relation algebra reduct (definition 5.40) of an n-dimensional cylindric algebra, or is a subalgebra of such a reduct, then we can arrange that fragments of a potential representation of up to n points are controlled by the cylindric algebra axioms of definition 5.16, just as smaller fragments (size four or less) are controlled by the axioms of relation algebras. The class of relation algebras that arise in this way is of course SfftaCAn (definition 5.45). On the face of it, the definition of Sg~aCAn is more abstract than that of RAn. For example, given even a finite relation algebra, it is not immediately clear how one might determine whether it is in SfftafA,, for given n. (In fact, for n > 5 this problem is undecidable: see theorem 18.13.) Maddux made progress towards a more concrete characterisation, by proving in [Madd78b, Madd89b] that any subalgebra of a relation algebra with an n-dimensional cylindric basis is in SgqaCAn. This is a useful sufficient condition for membership of SgqaCA,,. He established it by showing that an n-dimensional cylindric basis can be viewed as the atom structure of an n-dimensional cylindric algebra. The original relation algebra can be recovered from this cylindric algebra by taking its relation algebra reduct, and possibly then taking a subalgebra of that. Unfortunately, whether the condition is also necessary - - whether every algebra in sg~aCA,, is a subalgebra of a relation algebra with an n-dimensional cylindric basis - - remains an open question. So we cannot at present characterise S9~aCA,, by n-dimensional cylindric bases. Nonetheless, we will prove in chapter 13 that SfftaCA,, can be equivalently defined in the same way as RAn, using not cylindric bases but n-dimensional hyperbases. Hyperbases generalise cylindric bases by using hypernetworks instead of atomic networks. Hypernetworks contain more information than simply the relation algebra structure on n points of a potential representation. Each sequence of up to n points in an n-dimensional hypernetwork carries a 'hyperlabel'. The idea is that the points of the hypernetwork are controlled by an element of an n-dimensional cylindric algebra, and the additional hyperlabelling structure describes this n-ary relation more fully than the merely binary relation algebra structure on the points can do. We believe that this extra information is essential to capture sg~aCA,, and cannot be eliminated, but further research is needed.
Relativised representations How can we formalise the notion of 'potential representation' of a possibly nonrepresentable relation algebra? The answer is to use the relativised representations of definition 5.1. In chapter 5, we saw a natural way of defining a relativised representation for a relation algebra, or more generally a non-associative algebra,
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Introduction to approximations
A. We dropped the requirement that the representation be fully classical (i.e., the original unrelativised notion of representation), and instead allowed relativised interpretations of all the relation algebra operators. The unit (the interpretation in the representation of the top element 1) is a reflexive, symmetric relation on the base set M of the representation, and all operations, in particular negation ' - ' and composition ';', are relativised to the unit. So a pair (x, y) of elements of M stands in the relation r;s (where r, s E A) if there is z E M with (x,z) standing in the relation r and (z,y) in the relation s, so long as (x,y) is in the unit. The interpretation of ' - ' is defined similarly; the other operations are unaffected by relativisation since 1M is reflexive and symmetric. Maddux used this kind of relativised representation to characterise weakly associative algebras (section 5.1 and theorem 7.5). We now want to insist that 'on the small scale', the relativised representation does appear to be classical. We may then vary the scale up to which classical behaviour is required. If we only demand that things appear classical when considering at most three points of the base of the relativised representation, then we capture semi-associativity and hence the class SA. If we require classical behaviour of four-point fragments, we capture associativity and hence ordinary relation algebras. Indeed, these are alternative ways of defining SA and RA. Requiring classical behaviour for fragments of up to five points takes us to a strictly smaller class than RA, up to six takes us to a still smaller one, and so on. The 'limit' or intersection of these classes is RRA. Thus, we obtain a whole series of 'approximations' to, or perhaps 'n-variable analogues' of, RRA.
n-square relativised representations
In fact, there is more than one way of requiring classical behaviour of fragments of a relativised representation up to a given size. The basic requirement is that the relativisation (of negation and relational composition) is only visible when fragments of the representation above a certain size (say, n points) are considered. The n-square relativised representations of definition 5.7 are like this. Recall from definition 5.7 that a subset of a relativised representation M is a clique if all pairs of elements of it lie in the unit. We defined M to be n-square if for any clique C c_ M of size less than n, if x,y C C and (x,y) lies in the relation r ' s then there is a clique C' ~ C in M containing a point z realising this composition, so that (x,z) lies in the relation r, and (z,y) in s. An n-square representation is 'locally classical', in that if we look at it through a moveable 'window' only big enough to show n points, we will never discover using the relation algebra operations that it is relativised. n-fiat relativised representations A stronger requirement than n-squareness involves commuting quantifiers. To explain it, we will use 'clique-relativised' semantics. By definition 5.1, a relativised representation M of a non-associative algebra A is a model of a certain first-order theory RA in the signature L ( A ) of
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which each element of A is a binary relation symbol. Any n-variable formula r in the first-order language in signature L(A), written with variables x0,... ,X,_l, say, can be naturally interpreted in M by relativising quantifiers of q~ to cliques. In the clique-relativised semantics of r the range of every assignment of variables to elements of M used in the inductive evaluation of r should be a clique. Equivalently, we can syntactically relativise all quantifiers in q) to Ai,j
Unifying the three approaches So we have three approaches to approximating RRA - - by bases (RA,,), relation algebra reducts (SfftctCA,,), and relativised representations. In chapter 13, we will unify all of them. We show that for all finite n > 4: 9 sg~aD,, - Sg~aG,, - RA,, is the class of all relation algebras with an nsquare relativised representation, 9 sg~ctCA,, = S91a(CA,, n D,) = Sg~a(CA,, A G,,) is the class of subalgebras of relation algebras with an n-dimensional hyperbasis, and the class of relation algebras with an n-flat (and infinitarily n-flat, and n-smooth) relativised representation.
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Thus, in each case, the same class can be equivalently characterised in terms of bases, relation algebra reducts, and relativised representations.
The 3-dimensional case
Above, we required that n _> 4. The case n - 3 is rather special, because Sg~aG3 - WA ~ RA3 (cf. lemma 5.54) and S9~aCA3 ~ NA. The proof that a relation algebra reduct has a basis requires n _> 4. But all is not lost: we can show that RA3 - SA - Sg~a(G3 f-)CA3), this being the class of non-associative algebras with a 3-square and/or 3-flat and/or 3-smooth relativised representation.
The payoff We give some illustrations of the benefits of having this three-way characterisation of the classes RAn and Sg~aCAn.
Representation theory for RA Since RA = RA4 : Sg~aCA4, a corollary of our results is that the class RA of relation algebras is precisely the class of nonassociative algebras with 4-square (or equivalently, 4-flat or 4-smooth) relativised representations. Indeed, we will see in chapter 19 that the class of finite relation algebras is precisely the class of all finite non-associative algebras with finite 4square (or equivalently, finite 4-flat or 4-smooth) relativised representations. The representation theory developed here characterises arbitrary relation algebras, and may therefore be useful for studying relation algebras without classical representations. Bases and representability Using the representation theory developed here, we can give a direct argument (proposition 13.48) showing that the RAn (and the S91aCAn) converge to RRA. A similar argument using relational bases appears in [Madd91 b, p. 112]. Canonicity We can also prove that the varieties RAn (4 < n < to) are canonical by generalising Monk's theorem for RRA (theorem 3.36) to n-square (and n-flat) relativised representations. Axiomatising RAn and SfftaCAn It is easily seen that the varieties RAn and S~ctCAn (5 _< n < o~) are pseudo-universal classes (definition 9. l) by each of their characterisations. Being contained in RA, they are discriminator varieties. So we can use the techniques of chapter 9 to give equational axioms for them in three different ways, corresponding to our three kinds of characterisation. We do this in outline in theorem 13.55. Though the axiomatisations are infinite, and for n > 5 necessarily so, they are explicit and recursive. It may be interesting to compare the axiomatisations obtained from the three different characterisations of the classes.
Introduction to approximations
361
A hierarchy of approximations In part IV we will be better able to assess the merits of RAn and SfflaCAn as approximations to RRA. The following diagram and text summarises some of what we will find.
NA
RA3
RA4
uS
II
II
II
Sg~a(CA3 NG3)
S9~aG4
Sg~aG5
--.
II
Jl u
-..
S9~aG3
II WA
~f
SA
~f
3
RA
II
II
Sffta(CA3NG3)
SfftaCA4
3
RA5
3
SfftaCA5 3
...
-..
RRA
RRA
Inclusions between classes All classes shown in the diagram are canonical conjugated varieties, all except NA and WA are discriminator varieties, all inclusions are strict, and all except the ~ f s are not finitely axiomatisable (i.e., the smaller class, which may be RRA, is not finitely axiomatisable over the larger). Remark 15.13 will show that RAm SfftaCA,, for all finite m,n :> 5, so no more inclusions other than those derivable by transitivity can be added to the diagram above. Also, proposition 13.48 will show (combining work of Monk and Maddux) that
n RAn- n
3
Sg~aCA,,-RRA.
3
So by exercise 1 below, all inclusions derived by transitivity from the non-finitely axiomatisable ones shown in the diagram are also non-finitely axiomatisable. For example, RRA is not finitely axiomatisable over RAs. Lemma 5.54 and exercise 5.7(4) showed that Sg~aG3 - WA. The other equalities in the diagram will be proved in corollaries 13.47 and 13.50. It is well known that SA 3 RA. The inclusions RA4 3 RA5 3 -.. were proven strict in [Madd92]. Infinitely many of the inclusions SfftaCA4 _~ sg~aCA5 _~ ... were shown to be strict in [Madd89b], and in [HirHod ~02c] and chapter 15 it is shown that they are all strict. The non-finite axiomatisability results are proved in [HirHod00, HirHod01 a] and chapters 15 and 17. For n _> 6, RAn and Sg~aCA,, are not closed under completions (see section 17.7) and are therefore not Sahlqvist varieties (theorem 2.96). The analogous questions for n - 5 are open.
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362
In some ways, however, RAn is better behaved than SfftaCAn: 9 RA3 - SA, but as we saw in fact 5.46, sg~ctCA3 ~ NA. (This is why we use Sffta(CA3 A G3) in the bottom line of the diagram above. The irregularity caused is mostly only visual, since S f f t a C A , , - Sg~a(CAn A Gn) for finite n>4.) 9 Given finite n _> 5, it is decidable (in polynomial time) whether a finite relation algebra A is in RAn (corollary 12.32), but undecidable whether A E SfftaCA,, (theorem 18.13). 9 For any finite relation algebra A, we have A c RAn iff A has a finite n-square relativised representation (theorem 19.18). This 'finite model property' fails for SfftaCAn and n-flat relativised representations (proposition 19.19), although by theorem 19.20 it does hold for the class Sg~a{ C c CA,, : C finite}. 9 [Ste00, SteVen98] characterised RAn as the class of subalgebras of relation algebra reducts of 'Q,,-algebras', where Qn is an nZ-ary operator symbol similar to those defined by J6nsson in [J6n91], and those in section 8.2. The Qn-algebras form a finitely axiomatisable Sahlqvist variety (see exercise 12.4(12) below). We do not know of a similar subreduct characterisation of sg~aCA,,.
Approximations or analogues? The conclusion is that RAn and S~aCA,, do not approximate RRA very closely: for any finite n >_ 5, RRA and even RA,,+I are 'infinitely far away' from RAn in terms of axioms required to make up the difference, and similarly for Sg~aCA,,. Even Sg~nCAn is infinitely far from RAn. Perhaps these classes are better seen as finite-dimensional analogues of RRA. (The 'dimension' here ('n') is connected to the number of variables used in proofs in first-order logic; we will draw out this connection later.) They are non-finitely axiomatisable canonical varieties with many of the properties of RRA, and they have a workable representation theory.
Exercises 1. Let K,, (n _< co) be elementary classes of relation algebras (say), with Ko _~ Kl _~ ... and Ko~ : f~n