Equational Maps

EQUATIONAL MAPS W. FELSCHER Universitat Freiburg Let B and C be classes of non-empty abstract algebras and let @ be a map from B into C such that, for every BEB, the sets underlying B and @ ( B )are the same. 4 is called equational if there exist terms, formed with the operations of B, such that, for every BEB, the operations of 4 ( B ) are the same as the operations induced in B by these B-terms. If, moreover, 4 is bijective from B onto C and the inverse 4-l is equational from C into B, then 4 is called an equational equivalence. Equational equivalences were introduced in Malcev [I31 under the name of rational equivalences; equational maps occur in Lawvere [l I ] and Linton [I21 in the form of certain functors. Examples of equational maps abound : the map, converting every associative ring into a Lie-ring, is equational; the numerous possibilities to define Boolean algebras with varying kinds of operations give rise to equational equivalences, and some highly non-trivial examples of equational equivalences were exhibited by Cs6kany [3, 4, 51. The aim of the present article is the general study of equational maps and the investigation of how certain properties of a class B are reflected as properties of its image under an equational map. Also, some generalizations to relational systems and applications to syntactical transformations will be considered. The first three sections of this paper are preparatory. In Section 1 a general review of free algebras is given. All the facts stated are well known, although some of the proofs indicated may be new. Section 2 contains a detailed study of algebraic (or polynomial) operations on an algebra, useful later on in Section 5. The procedure followed here is influenced partly by Schmidt [16]. However, the applications intended make it necessary to delve somewhat deeper into the properties of n-ary operations induced by m-ary terms where n < m. The short Section 3 assembles some simple facts about reducts. The fourth section begins with a precise definition of equational and functorial maps. The latter ones are those maps from a class B into a class C 121

122

W. FELSCHER

which, in addition to preserving underlying sets, also have the following property: for any two algebras B and G of B, the homomorphisms from B into G are the same as the homomorphisms from 4 ( B ) into +(G). Every equational map is functorial. Conversely, if B contains free algebras with sufficiently many generators, then every functorial map is equational (Theorem 1). In particular, this yields a theorem of Malcev [13], stating that functorial equivalences between quasi-primitive classes are equational. In Section 5 the construction of equational maps from given sequences of terms is studied. It follows that every equational map $r from B into C can be extended to an equational map, defined by the same terms as 4, from Es(B) into Es(C), where Es(B) is the strict equational closure of B (Theorem 3). If B is strictly equational (i.e. B=Es(B)) and 4 is an equational equivalence from B onto C, then C is strictly equational (Corollary 1). - Let 4 be an equational map from B into C. It follows from Lemma 9 that a function g - called reductive - can be found which transforms every term in operations of C into a term in operations of B; moreover, g is recursive in the sense that it is a homomorphism between certain algebras of terms. Now let 4 be an equational equivalence from B onto C and let B be defined by a set M of equations; let the transformation g be determined with respect to 4-l. Then C can be defined by the equations which arise from M under g, together with another set of equations saying that g transforms the B-terms, defining 4, into terms corresponding to the basic operations of C (Theorem 4). - Finally, it is shown that every class B is equationally equivalent to a class C such that the arities of basic operations of C are cardinal numbers and are minimal for this situation (Theorem 5). The sixth section begins with an outline of the relationship between certain infinitary languages and their models, namely relational systems with not necessarily finitary relations. The only non-trivial statement here concerns the existence of general substitutions ; the rather technical proof has been placed in an appendix. Imitating the structure of the definitions in Section 4, it is obvious how t o introduce definable maps 4 between classes B and C of models; if open formulas only are admitted, also open maps may be considered. A closer inspection of Section 5 then shows that the principal definitions and the theorems there can be translated almost verbally to this new situation. In this connection, it is the author’s pleasure to acknowledge his indebtedness to H. J. Hoehnke. In July 1966, the author had occasion to discuss an earlier draft of the first five paragraphs with Hoehnke who, in [S] had studied first-order definable equivalences (‘Strukturaquivalenzen’). Hoehnke then observed that the extension theorem (Theorem 3) could be

EQUATIONAL MAPS

123

generalized to give an extension theorem for these definable equivalences, and he provided a proof along the lines of his article [8]. It was only then that it became clear to the author, that the arrangement of Section 5 was already such that the concepts ‘algebra’ and ‘equation’ only had to be replaced by ‘model’ and ‘bi-implication’in order to handle the more generalcase. There is, finally, a second part of Section 6 which deals with syntactical transformations. These are nothing but reductive functions belonging to certain equational equivalences, namely equivalences varying the structure of the class of full cylindric set algebras. The result then is that syntactical transformations preserve axiomatizations: if syntactical transformations between languages P L 1 ,P L 2 are given, then an axiomatization of either tautologies or the consequence operator of PL’ is, in a natural way, transformed into a corresponding axiomatization for P L 2 , and if the first axiomatization is finitary then so is the second.

1. A review of free algebras In the following, the usual notions of set theory will be employed. If Xis a set, then % X shall be the set of all subsets of X , and if X , Yare sets then X y shall be the set of all functions from Y into X . A function f with domain Y will also be written as a sequence ( f ( y ) l y e Y ) . 1ffe.Y’ and 2s Y then f 12 shall be the restriction off onto 2, and f * (2)shall be the image of 2 underf. Ordinal numbers will be construed as sets such that every ordinal number n is the set of all ordinal numbers m smaller than n . In particular, the empty set 0 is the smallest ordinal number, and w is the smallest infinite ordinal number. Cardinal numbers will be identified with initial numbers, i.e. ordinal numbers that are not equipotent with any smaller ordinal number. If E is a set then card(E) shall be the cardinal number of E. A cardinal number n is regular if it cannot be decomposed into a union of less than n sets of cardinality less than n. If E and n are sets, an operation of arity n on E is a function from E ninto E. A type A is a sequence ( n , l i e I ) of sets where I is a non-empty set. A type is said to contain constants if some of the sets n, are empty. One defines rank(A) as the smallest infinite cardinal number m such that card(n,),
124

W. FELSCHER

of arity ni on E ; the underlying set E is called the carrier of A and is denoted by u(A). An algebra is called finitary if its type is finitary. An algebra is called empty if its carrier is empty; it is called singular if its carrier contains at most one element. A class of algebras is called singular if each of its elements is singular. All classes of algebras considered in the sequel are assumed to be not empty. Iffinitary algebras only are considered, none of the proofs given in this article will use the axiom of choice ( A C ) . From now on, in this paragraph algebras of a fixed type A will be considered. The definition of homomorphism, subalgebra, congruence relation, quotient algebra and of a product of algebras is obvious (for a detailed description cf. Slominski [20], Schmidt [I61 and, for the finitary case, also Cohn [2]). For algebras A , B the set of all homomorphisms from A into B will be written as Hom(A, B ) . If A is an algebra and X c u ( A ) then the closure [XI (or [ X I Aif necessary) of X shall be the intersection of all u(B) such that X c u ( B ) for subalgebras B of A ; the uniquely determined subalgebra C of A such that u ( C ) = [XI then is called the subalgebra generated by X in A. X is called closed in A if X = [XI. If Xc Y c u ( A ) and Y is closed, then [ X ] E Y ; this trivial but useful observation is called the principle of algebraic induction (Schmidt [16]).As an application, one finds that two homomorphisms from A into an algebra B coincide on [XI provided they coincide on X . Further, making use of the regularity of dim(A), algebraic induction yields for every a e [ X ] a subset X,EX such that a ~ [ x , and ] card(X,)
125

EQUATIONAL MAPS

equations. An equation ( u , v) holds in an algebra A if h(u)= h(v) for every hEHom(P, A). If A is a class of algebras, then Q ( A ) - or Q(A, X ) if necessary - shall be the set of equations holding in every A E A . Q(A) is a congruence relation on P since it is the intersection of all congruence relations induced by homomorphisms hEHom(P, A ) for A E A . Defining Q ( A ) = Q ( { A } ) for algebras A , one obtains that Q(A) is also the intersection of all Q ( A ) for A E A . If ( u , D ) E Q ( A )then, for any gEHom(P,P), also { g (u),g (v)) E (?(A)- for if A E Aand h E Hom(P, A ) then also hg EHom(P, A). Let X contain more than one element; a class A then is singular if and only if Q ( A ) contains an equation {x, y ) such that x,y in X,x # y ; in that case Q ( A ) = u ( P ) x u ( P ) . Let Y be a subset of X and let Py be the subalgebra generated by Y in P. Obviously, Py is absolutely freely generated by Y. For every class A of algebras one obtains Q ( A , Y ) = Q ( A , X ) n ( u ( P , ) x u(Py)). For if A E A and heHom(P, A ) then h /u(Py)EHom(Py,A ) , which implies Q ( A , Y)E Q ( A , X ) . If, on the other hand, kEHom(P,, A ) , i,b = k Y, choose cp~u(A)' such that cp extends i,b. If hEHom(P, A ) extends cp, then h ru(Py) = k ; thus Q ( A , X ) n ( u ( P , ) x u ( P , ) ) s Q ( A , Y ) . Let P be absolutely freely generated by X . I f a class A contains an algebra A , A-freely generated by a set Z equipotent with X , then A is isomorphic to P/Q(A). For let p be a bijection from X onto Z , let pEHom(P, A) be the extension of p, let p = hn be the decomposition into a natural epimorphism from P onto a quotient algebra P/Q and an isomorphism h E H o m ( P / Q , A ) . It will be sufficient to show Q = Q(A). Now A E A implies Q ( A )c Q. If, on the other hand, BEA andjEHom(P, B), define x=j 1X.Since A is A-freely generated by Z , x-p-' extends to kEHom(A, B ) ; as j and kp coincide on X, one finds j = kp = khn. Thus Q E Q (A). Conversely, let P be absolutely freely generated by X , let A be any class of algebras, and let n be the natural epimorphism from P onto P / Q ( A ) .Any map x from n * ( X ) into the carrier u(A) of an algebra A E A determines a kEHom(P, A ) which extends X-(TC 1X ) . Since A E A , k factors through x, i.e. k=hn where hEHom(P/Q(A),A ) . Since n*(X)generates P / Q ( A ) ,h is the unique extension of x in Hom(P/Q(A),A ) . Thus every algebra in A , isomorphic to P / Q ( A ) ,is A-freely generated by a set equipotent with n* ( X ) . Precisely in case A is not singular, n 1X is a bijection, i.e. n* ( X ) equipotent with X . For every class A of algebras define S ( A ) , H ( A ) , I(A) as the class of all subalgebras, homomorphic images and isomorphic images of algebras of A respectively; define P(A) as the class of all algebras which are products of arbitrary families of algebras in A. A class is called primitive if it is closed

r

126

W. EELSCHER

with respect to S, H and P; it is called quasi-primitive if it is closed with respect to S, I and P. If A is closed with respect to S and P, then the axiom of choice ensures that I(A) is quasi-primitive. Let P be absolutely freely generated by X and assume that P is not empty. Then, for any class A, the algebra P/Q(A) belongs to ISPIS(A). For let Q be the set of all congruence relations on P, induced by homomorphisms into ). R belongs to algebras of A; let R be the product ~ ( P / Q ~ Q E Q Then PIS(A) since each of the algebras P/Q belongs t o IS (A). Let T , be the natural epimorphism from P onto P/Q; let p a be the epimorphic projection from R onto P / Q ; letpEHom(P, R) be defined by nQ=pQ *pfor Q E Q ; letp=hn be the decomposition into a natural epimorphism from P onto an algebra PIS and a monomorphism into R . Since nQ=pQ.h.n for every QEQ, one obtains that S = ~ ( Q ~ Q E Q )S=Q(A). , Thus P/Q(A) is isomorphic to a subalgebra of R . Let A be a non-singular, quasi-primitive class of algebras. Then A contains, for any set X , an algebra A , A-freely generated by X . For let P be absolutely freely generated by X. If P is not empty, P/Q(A) belongs to A, and a familiar replacement construction produces an algebra A , containing X , and an isomorphism from P/Q(A) onto A which sends T*(X) onto X . If P is empty, also X is empty and the type A does not contain constants. In that case, the empty algebra is a subalgebra of any algebra of A and may be chosen for A . - If closedness with respect to S is replaced by closedness with respect to non-empty subalgebras, then the assertion still holds for any non-empty set X . A class A of algebras is called equational if there exist a set X such that card(X)=rank(A), an algebra P absolutely freely generated by X , and a set A4 of A-X-P-equations such that A consists exactly of those algebras in which the equations of M hold. In that case, A is called deJned by the set M . Obviously, this definition depends only on the cardinality of X but not on the actual set X nor on the way in which the particular algebra P may have been chosen. If A is equational, then A is defined by the set Q(A, X ) of A-X-P-equations. If A is arbitrary, let E(A) be the class of algebras defined by the set Q (A, X ) for some X such that card(X) = rank(d). Then Q (A, X ) = Q(E(A), X ) , and E(A) is the smallest equational class containing A; E(A) is called the equational closure of A, A class A is called strictly equational if A consists exactly of the non-empty algebras in which, for some X , P and M as before, the equations of M hold. Correspondingly, for a class A of nonempty algebras the strict equational closure Es(A) is defined as the class of all non-emPtY algebras of E(A).

EQU.4TIONAL MAPS

127

Let P be absolutely freely generated by a set Y and assume that P is not empty. Any A-Y-P-equation, holding in an algebra A , holds a fortiori in any subalgebra of A . Any equation, holding in every factor A, of a product R = n ( A , J s € S ) , also holds in R since operations in R are performed coordinatewise. Finally, it is easy to see that any equation holding in an algebra A also holds in every homomorphic image of A ; if A is finitary, then the axiom of choice is not needed here. Consequently, every equational class is primitive. If A is arbitrary, then P/Q(A, Y) belongs to ISPIS(A); hence the equations from &(A, Y ) hold in P/Q(A, Y). LEMMA 1. Let A be a class of algebras, let P be absolutely freely generated by a set Y, let n y be the natural epimorphism from P onto P/Q (A, Y ) . Then P/Q(A, Y ) is E(A)-freely generated by n;(Y). For the proof, it can be assumed that P is not empty. Then P/&(A, Y ) belongs to ISPIS(A) and, therefore, to E(A). Let X be a set such that card(X)=rank(A). Since Q(A, X)= &(E(A), X),the algebra P/Q(A, X) is E(A)-freely generated by nz(X).If the lemma is true for a certain set Y , it will also be true for any set equipotent with Y. It will be shown below that (a) if the lemma is true for a set Y then it is true for every subset Z of Y; (b) if the lemma is true for a set 2 such that card(Z)=rank(d) then it is true for every set Y such that ZE Y. This will finish the proof, for the lemma has been shown for the set X ; if Y is arbitrary, it will be true for X x Y by (b) (provided Y is not empty) and, therefore, for Y by (a). So let Z be a subset of Y and let Pz be the subalgebra generated by Z in P.In order to deal with (a), one has to prove that Pz/Q(A, Z ) is E(A)freely generated by n f ( Z ) . This will be done if, for any G€E(A), every k€Hom(P,, G) factors through nz. If k is given, let cp be an arbitrary extension of k 12 in u(G)' and let hEHom(P, G) be the extension of cp. By assumption, the lemma holds for Y ;hence h factors through ny.Therefore h identifies the equations of Q(A, Y ) ; since &(A, Z)=Q(A, Y)n(u(Pz) xu(Pz)) and k = h ru(Pz), also k identifies the equations of Q(A, 2). In order to prove (b), let 2 and Y be given such that Y z Z , card(Z)= rank(A); it can be assumed that card(Y)>card(Z). It will be sufficient to show that, for any GcE(A), every hEHom(P, G) factors through ny. Let ( u , v) be an equation in &(A, Y); there exist subsets Y,, Y, of Y such that U E [YJ, v ~ [ Y , l ,card(Y,)
128

W. FELSCHER

card(Z)< card( Y ) , it follows from (AC) that there exists W' G Y such that card(W')=card(W), W ' n W = W ' n Z = O . Sending first W into W' and then W' into Z, a bijection y of Y onto itself may be chosen which maps W into Z . Let g be the automorphism of P which extends y. Then h will identify (u , u ) if and only if h - g - ' identifies ( g ( u ) , g ( v ) ) . But with ( u , u ) also (g ( u ) ,g ( u ) ) belongs to Q(A, Y ) ; since g * ( W ) c Z , g * ( [ W ] ) c u ( P , ) , the equation (g (u), g ( u ) ) even belongs to Q(A, Y )n(u(Pz)x u(Pz))= Q(A, Z ) . By assumption, the lemma holds for 2 ; hence k = h . g - l ru(P,) factors through n,. Consequently, h v g - l as well as k identifies (g(u),g ( u ) ) . - If only finitary algebras are considered, the bijection y can be constructed without any use of (AC). For in that case rank(d)=dim(A)=w shows that the set W can be found such that card( W )
2. Algebraic operations In this section algebras of a fixed type A will be considered.

EQUATIONAL MAPS

129

Let E and X be sets and denote by S(E, X ) the power of E with exponent E X , i.e. the set of all operations of arity X on E. If Y c X , let r," be the injection of S ( E , Y ) into S ( E , X ) such that, for any dES(E, Y) and any cp~E', one has r:(d) (cp)=d(cp 1 Y ) . Let A be a non-empty algebra and let X be a set. Then Op(A, X ) shall be thepower o f A with exponent #(A)', i.e. the algebra with carrier S(u(A), X) and operations defined coordinatewise. If c p ~ u ( A )let ~ , nf be the epimorphic projection from O p ( A , X ) onto A at cp, i.e. n:(d)=d(rp) for any deu(Op(A,X)). If Y G X , then ry" becomes a monomorphism ryAX from Up(A, Y ) into @ ( A , X ) which, occasionally, will be written simply as ry". If XEX, define e:' in u ( O p ( A , X ) ) by e$"(cp)=cp(x) for every cp~u(A)'. If Xf 0, let H ( A , X ) be the subalgebra of O p ( A , X )generated by (e:"lxeX}. If y e Y and Y G X , then r:(eyAY)=efX. Hence r: ru(H(A, Y ) ) is a monomorphism from H ( A , Y ) into H ( A , X ) which, again, will be written as r:' or r,". Let A be as before, XfO, and let P be absolutely freely generated by X. One defines an epimorphism e, from P onto H ( A , X ) such that e, (x)= e,"' for XEX.Let cp be in u(A)' and let h , be the extension of cp in Hom(P, A). Then nf * e, = h,, since these homomorphisms coincide on X . Therefore e A ( r ) ( c p ) = h g ( r )holds for any r e u ( P ) . The congruence relation on P induced by eA is the set Q(A)= Q ( { A ) ,X ) of all A-X-P-equations holding in A , because eA(u)=eA(v)if and only if h,(u)=h,(v) for every q ~ u ( A ) ' (cf. Schmidt [17], Satz 8). Hence there exists an isomorphism h, from P / Q ( A ) onto H ( A , X) such that h,n,=e,, where nA is the natural epimorphism from P onto P/Q(A). Assume now that A is not singular. Then H ( A , X) is E({A})-freely generated by {e$"lxEX} because hA(nA(x))=e,"'. In particular, E({A}) contains each of the algebras Op(A, Y ) and H ( A , Y ) for arbitrary Y, since it is closed with respect to products and subalgebras. Let A , X and P be as before. Since the equations of Q ( A ) hold in P / Q ( A ) , one has Q(A)=Q(P/Q(A))=(z(H(A,X ) ) . Thus there exists an isomorphism h, from P / Q ( A ) onto H ( H ( A , X ) , X) such that hHn,=eH(,,.). Defining hX=hH*hA1,one obtains an isomorphism from H ( A , X) onto H ( H ( A , X ) , X ) which, for every r e u ( P ) , maps e A ( r )onto eH(,,')(r); in particular, e:' is mapped onto eff(A3X),x. Now let g be in u ( H ( A , X ) ) , let q be in u(H(A, X))', and let cp be in U ( A ) ~Then . n,"(h,(g) (q))=g(nf.q) holds. For let r be in u ( P ) such that g = e A ( r ) and, therefore, hX(g)=eH(,,')(r). Define i = n f . q , a n d leth',betheextensionofq in Hom(P,H(A,X)).Theng(nf*q)= e A ( r )(A)=h,(r) and n~(e,(,,xt(r) ( q ) ) = n f ( h : ( r ) ) holds. But nf.h: extends A in Hom(P,A); thus h,=nf.h,' and hA(r)=nt(h,'(r)).- The equality

130

W. FELSCHER

n:(hx(g)(u]))=g(n:*u]) may be written as (h,(g)(y))(cp)=g((u](x)(cp)lxEX)). Thus h,(g) is the power with exponent X of g and h,(g) ( u ] ) is the superposition of g and y. Let A be a non-empty algebra, Y c X , X#O. Let Op(A, X;Y ) be the image under r," of Op(A, Y ) in O p ( A , X ) . Then d~u(Op(A,X)) belongs to u(Op(A, X;Y ) ) if and only if, for all cp, x in u(A)', cp 1 Y=x 1 Y implies n:(d)=n:(d). Assume now Y#O and let H ( A , X;Y ) be the image under r; of H ( A , Y ) in H ( A , X).Then H ( A , X ; Y ) is the subalgebra generated by { e e x ( y EY ) in H ( A , x). LEMMA 2. u ( H ( A , X ; Y ) ) = u ( H ( A ,X ) ) n u ( O p ( A ,X;Y ) ) . This is obvious in case A is singular; if A is not singular, still the set on the left side is contained in the set on the right side. Observe that H ( A , X ) is E({A))-freely generated by { e t x [ x E X }and that H ( A , Y ) belongs to E({A)). Let o be a map from X onto Y such that 0 Y is the identity; let f be the epimorphic extension of o from H ( A , X)into H ( A , Y ) such that f (e:') = e::). For every $ in u(A)' one finds that ni*f=n& since both these homomorphisms map e t x onto $(o(x)).Let d be in u ( H ( A , X))and assume that cp 1 Y = x Y implies n:(d)=n;(d) for all cp, x in U ( A ) ~it; will be sufficient to show that d = r : ( f ( d ) ) . But with $=cp Y one finds n $ ( r , " ( f ( d ) ) ) = f (d))=n;.,(d)=n;(d) for every cp in u(A)', because $so 1 Y = ((q Y ) . o ) 1 Y = q Y due to the choice of o. Let A be a non-empty algebra and let X be a non-empty set. A subalgebra H ( A , X;0) of H ( A , X) can be defined by u ( H ( A , X ;O))=u(H(A, X))n u(Op(A, X ; 0)). Let V , Y be sets such that Vc Y G X , Y#O. Then r," determines an isomorphism from H ( A , Y ; V ) onto H ( A , X ; V ) . For it is clear that H ( A , Y ; V ) is mapped into H ( A , X;V ) . Conversely, Lemma 2 gives u ( H ( A , X ; V ) )c u ( H ( A , X ; Y ) ) because u ( H ( A , X ; V ) )= u ( H ( A , X))n u(Op(A,X ;V ) ) c u ( H ( A ,X ) ) n u ( O p ( A , X ; Y ) ) = u ( H ( A ,X ; Y ) ) .Therefore , )) for any d € u ( H ( A ,X ; V ) ) there exists d ' € u ( H ( A , Y ) ) and d " ~ u ( O p ( A V such that d= r,"(d') = r,"(d"). Then r,"(d') = r," (d")= r f r ; ( d " ) implies d'=r;(d") and d ' e u ( H ( A , Y ; V ) ) . Let A be a non-empty algebra, let X be a non-empty set. If Z is not empty, then r$"' is an isomorphism from H ( A , X;0) onto H ( A , X u Z ; 0), and r:"' is an isomorphism from H ( A , Z ; 0) onto H ( A , XuZ;0). Therefore the counterimage under r t of H ( A , X;0) in Op(A, 0) does not depend on X . Define as H ( A , 0) the subalgebra of Op(A, 0 ) which is this counterimage. Let $ be the unique element in u(A)O; then T$ is an isomorphism of O p ( A , 0) onto A which maps H ( A , 0) onto a subalgebra C ( A ) of A. Now $=cp 10

r

.I( r

EQUATIONAL MAPS

131

for any X and any cp~u(A)';further n$= n;.rz. Therefore ng determines an isomorphism, not depending on cp, of H ( A , X;0) onto C(A). In general, rcf maps H ( A , X ) onto the subalgebra of A , generated by the range cp*(X) of cp, because n,"maps the generating set { e t x l x E X } of H ( A , X ) onto 'p* (X) (cf. Schmidt [16], Theorem 5). If B is a non-empty subalgebra of A , a sequence cp can be found in u(B)'; hence [ c p * ( X ) ] ~ u ( B shows ) that n: maps H ( A , X ) into B. Thus B contains the image of H ( A , X ; 0) under nf, i.e. C(A) is contained in every non-empty subalgebra B of A (cf. Schmidt [18], Theorem 4). Consequently, if C ( A ) is not empty, it is the smallest non-empty subalgebra of A . This holds in particular if the type A contains constants ; in that case, C ( A ) is generated in A by the empty set and, therefore, the isomorphic algebra H ( A , 0) is generated in Op(A, 0) by the empty set as well. - The elements of H ( A , X ) are called algebraic (or polynomial) operations of arity X on A ; in particular, the elements of H ( A , 0) are the constant algebraic operations, and the elements of C ( A ) are called algebraic (or equationally definable) constants of A . A first mathematical definition of algebraic operations was given by McKinsey-Tarski [ 141; the present treatment is influenced by Schmidt [16]. If X # 0 or if the type A contains constants, then the algebra H ( A , X ) is the same as Schmidt's H X ( A ) ;if A does not contain constants, H o ( A )will be empty while H ( A , 0) need not to be so. Let A be a non-empty algebra and Iet X be a non-empty set. If d € u ( H ( A , X ) ) , define Supp(d) to be the set of all Y such that YEX, d € u ( H ( A , X;Y ) ) . Then Supp(d) is a filter: if Y c Z c X , Y~ Supp(cl), then ZESupp(d); if YESupp(d), Z ~ S u p p ( d )then , Y n Z E S u p p ( d ) . Further, if X E Z and d € u ( H ( A ,X ) ) , then Supp(d) is a base for the filter Supp(rg(d)). Since any d € u ( H ( A ,X)) belongs already to a subalgebra H ( A , X ; Y ) such that c a r d ( Y ) < d m ( d ) , every filter Supp(d) has a base of sets of cardinality less than dim(A). In particular, if the type under consideration is finitary, then Supp(d) has a base of finite sets and, therefore, contains a smallest set. However, there are examples when Supp(d) will not contain a smallest set. For let X , 2 be infinite sets, Z G X , Z # X , and let a, b be different elements of X . For every cp E Xx,define d by d(cp)= a if 'p* (A') n Z is finite, d(q)=b otherwise. If an algebra A can be found such that d e u ( H ( A , X)), then Supp(d) will contain precisely the sets Y such that X - Y is finite. Now define A = ( X ) , A = ( X , ( d ) ) ; then O p ( A , X ) is an algebra ( X X , d ' ) . Define x€(XX)' by x(x)=e$X for X E X ; d € u ( H ( A , X ) ) will be shown if d=d'(x). But for every VEX' one has n:.x='p, whence nt(d)=d(cp)= d(n; .x)= rcf (d' (x)) since rcf is homomorphic. Let A be a non-empty algebra and let P be absolutely freely generated by

132

W. FELSCHER

a non-empty set X.If Y c X , let G ( Y )be the set of all endomorphisms g of P such that g 1 Y is the identity and g 1X maps X into X . If r ~ u ( P )the , fact that YESupp(e,(r)) can be expressed with help of equations which do not depend on the particular algebra A : LEMMA 3. If YZO, then Y~Supp(e,(r))if and only if, for every g E G ( Y ) , the equation ( r , g ( r ) ) holds in A . If, moreover, card(X)>dim(d) then this equivalence remains true also for Y = 0. )), If q = g YX and For a proof, assume first Y ~ S u p p ( e ~ ( r gEG(Y). cp~u(A)' then cp and cp-q coincide on Y ; hence h,(r)=e,(r)(cp)= e,(r)(cp.q)=h,.,(r)=h,(g(r)). Assume now that the given equations hold in A . If Y#O, let q be a map from Xonto Y such that q Y is the identity, and let gEG( Y )be the extension of q . If cp, $ are in U ( A )and ~ coincide on Y , then cp.q=$.q and, therefore, eA(r)(cp)=h,(r)=h,(g(r))=h,.,(r)=h,.,(r)= h,(g(r))=h,(r)=e,(r)($). In order to cover the case Y=O, assume now that card(X)>dim(d). Let W c X be a set such that e,(r)Eu(H(A, X ; W ) ) and card(W)
n

LEMMA 4. Supp(d)=

n( S u p p ( p g ( d ) ) l B ~ B ) .

EQUATIONAL MAPS

133

For a proof, let Y be in u ( P ) such that e,(r)=dand, therefore, eB(r)=pz(d) for BEB. Since Q(A) = Q(B), it follows from Lemma 3 that both sides of the assertion contain the same non-empty sets. Assume next that X contains at least two elements. Then there exist non-empty subsets Y, Z of X such that Y n Z = O . If 0 belongs to every Supp(pi(d)), then so do Y, 2. Hence Y , 2 belong to Supp(d). But since Supp(d) is a filter, also 0 belongs to Supp(d). Conversely, O ~ S u p p ( d )implies O ~ S u p p ( p i ( d ) for ) every B E B . Assume, finally, that X contains only one element, and let 2 be a set such that X G 2, X Z Z . For any B E B the homomorphisms r p . p g and p g . r i z from H ( A , X) into H ( B , Z ) are equal since they coincide on the generators e:', X E X . Since the lemma has been proved for r:"(d), one obtains Supp(r:"(d))= ( s u p p ( p ~ ( r , ~ ~ ( d ) )= ) i ~ €( s ~u ) pp(r,~~(p,~(d)))i~ since E ~ ) .s u p m is a base for the filter Supp(riZ(d)),one has for every Y GXthat Y ~ S u p p ( d ) if and only if Y ~ S u p p ( r ; " ( d ) ) ;likewise, Y ~ S u p p ( p g ( d )is) equivalent to Y ESupp (rp( p i ( d ) ) )for any B EB. Let A be a non-empty algebra, let X , Y be sets, Y c X , X Z O . Let P be absolutely freely generated by X and let Py be the subalgebra of P, absolutely freely generated by Y. If YfO then H ( A , Y ) is isomorphic to Py/Q(A, Y ) . Let e: be the epimorphism from Py onto H ( A , Y ) ; then rF-eI=e, ru(Py) since both these homomorphisms coincide on Y. (it may be remarked that the isomorphism Y; from H ( A , Y ) onto H ( A , X;Y ) corresponds to the isomorphism from P,/Q(A, Y ) onto (eA1eA)*(Py)/Q(A,X ) , given by one of the so-called theorems of isomorphism.) If Y=O, assume that Po is not empty, i.e. that the type A contains constants. Then H ( A , 0) as well as the isomorphic algebra H ( A , X ; 0) is generated by 0; thus eA maps Po onto H ( A , X;0). Combining eA ru(Po)with the isomorphism from H ( A , X;0) onto H ( A , 0), one obtains an epimorphism el: from Po onto H(A,O) such that, again, rt*e;=e, ru(Po). This may be read as saying that, if there are constant names for some elements d of u ( H ( A ,0)) (i.e. elements r€u(P0) such that e,(r)=d), then there are such names for all elements of u ( H ( A , 0)) (a remark to this effect also in Lawvere [I I], Chap. 11, prop. 5). In any case, if e: is defined then the equality r:.e:=e, ru(Py)implies that, for r ~ u ( P ) , one has YESupp(eA(r))if and only if there exists S E U ( P ~such ) that ( r , S) holds in A . Now let B again be a non-singular class of algebras. With a suitable algebra A - e.g. P / Q ( B , X ) - Lemma 4 then gives

n

n

LEMMA 5. Assume that Y f O or u(Po)#O. If r ~ u ( Pthen ) Y E n (Supp(e,(r))lBEB) if and only if there existssEu(Py) such that ( r , s) holds in every BEB.

134

W. FELSCHER

As an immediate consequence, one obtains the following: Assume that Y # 0 or u(P,) # 0; let r be in u(P). If for every BEB an element u,eu(Py) can be found such that ( r , u , ) E Q ( B ) , then a uniform ueu(Py) can be chosen such that ( r , u ) E Q ( B ) for every B E B . Let X and n be sets such that there exists an injection p from n into X . If E i s a set, then p determines a bijectionj, from S(E, n) onto S(E, D*(n)) and an injection r ~ = r & , , . j , from S(E, n) into S ( E , X ) . Now let A = ( n , l i ~ I ) be the type under consideration and let X be such that card(X) 2 rank(d). A A-coordinate system f o r X shall be a pair (K, (piliEZ)) such that K is a class of injections into X , { p i l i E Z } c K and every pi is a (distinguished) injection from ni into X . If the type A is ordinal, it will be sufficient to demand that K is a set, containing an injection p, for every ordinal n such that card@) j n i , r:. Assume now that a fixed A-coordinate system for X is given. Let P = ( u ( P ) , ( f i l i ~ I ) be ) an algebra, absolutely freely generated by X , and let A = (u(A), (fiAli~1))be a non-empty algebra. For every p,, the map r,U(,), restricted to u ( H ( A ,n)), becomes a monomorphism r,” from H ( A , n) into H ( A , X ) ; r t will be written instead of r,$ For every c p e ~ ( A )every ~ , ieIand every AEX”’ one obtains e A ( f , ( l . ) )( 4 0 ) = S , ~ ( q a A ) because fi”(cp.A)= h , ( f i ( l ) ) since h , is homomorphic. In particular e A ( h ( P i ) )(q)=f!(cp.p,) and therefore, p* (n,)E Supp (e, (f i (pi))). Further, one finds eA( f , (Bi)) = rA (A”) since rA (LA)( ~=1(‘;W.Cn,, *ji(AA>) (CP) = (ji(LA)> (40 t P* (nil) = LA((9iB* (ni)).Pi>=fi” (9.Bi). Retaining the last notations, let r be in u ( P ) and let p, be in K such that f i * ( n ) ~ S u p p ( e , ( r ) )i.e. , e,(r)Eu(H(A, X ; p*(n))). The unique element rA,fln in u ( H ( A , n)) such that r,” (r = e, ( r ) shall be called the operation ofarity n on A defined by r. If no ambiguities can arise, r A S nwill be written instead of rASfln. In particular, (J;:(j?i))A~”i=A” for every i E 1 . If G is another algebra and geHom(A, G) then ghq=hg,, for any c p ~ u ( A )hence ~ ; ( g . e , ( r ) ) (q)= g ( e , ( r ) ( q ) ) = e , ( r ) (g’cp) for every r E u ( P ) (cf. Schmidt [17], Cor. of Satz 5). If, in addition, p*(n)ESupp(e,(r)) and p*(n)ESupp(e,(r)) then s ( r A , ” ( $ ) ) = r G , ” ( g - $ )for every $EU(A)”. For choose c p ~ u ( A )such ~ (cp)=(r&,,-jn(rA,”))(cp)= that cp rp*(n)=$.p-,’ ; then eA(r)(q)=(r,”(rA”’)) r”,”((cp rP*(n)).Pn)=rA,’($)and, in the same way, eG(r)( g * c p ) = r G 7 n ( g . $ ) . -If an equation ( r , s) holds in a n algebra A then Supp(e,(r))= Supp(e,(s))

EQUATIONAL MAPS

135

and, for any n such that P*(n)ESupp(eA(r)),also r A ~ " = s A ~If,' . conversely, B * ( n ) ~ S u p p ( e ~ ( r ) ) n S u ~ p ( eand ~ ( sr)A) r n = s A 2 *then , ) = g ( ( s ( Y )(cp t Z ) l Y E y>>= [ g : sl (cp YZ). Thus r ; [ g : q ] belongs to u ( H ( A , X;Z ) ) . Then [g: q l ~ u ( H ( A2)) , follows from Lemma 2 and, in case Z=O, from the definition of H ( A , 0). (For a different proof cf. Schmidt [16], Cor. 3 of Th. 20.) The following remarks shall establish some connections with clones in the sense of Ph. Hall (cf. Cohn [2]);they will not be needed in the later paragraphs. Let E and X be non-empty sets and let X be chosen such that card(X) is an infinite regular cardinal number. For every set Y and any Y E Y define ef' in S(E, Y ) by e,"'(cp)=cp(y) for any ~ E E ' . A subset M of U ( S ( E , Y ) I Y c X and card(Y)
u

136

W. FELSCHER

on I,and let A be the algebra ( E , ( L " l i ~ 1 ) )IfLAEM, . sayL"ES(E, n), define x ~ u ( H ( A , n ) ) "by x(k)=e$" for k
f i H ( A 9 n )which ( ~ ) , shows that Miscontainedin U(u(H(A,m))lm< c a r d ( X ) ) .

On the other hand, if m>O then M contains the generators e t m ,kO the inclusion u ( H ( A , m))EMfollows by algebraic induction from the fact that M is closed with respect to superposition. Finally, u ( H ( A , 0)) E M follows from u(H(A, 1)) 5 M and the property (iii) of clones. If A is an algebra of type A and X is a set such that dinz(d)max(dim(A'), d i m ( d 2 ) ) ;if both types are ordinal, let fi be a bijection from card(X) onto X . Then u(H(A', X ) ) = u ( H ( A 2 ,X ) ) holds ifand only if A', A2 determine the same non-ordinal or, in case both types are ordinal, the same ordinal E-X-clones (cf. the proof of Lemma 7 below).

3. Reducts

In this section two types A = ( p j l j € J ) and A' = ( n i l i € l )will be considered, and dl shall be a reduct of A :there shall exist an injection z from Z into J such that p7(i,=ni for i E Z - in most cases Z will be a subset of J and z will be the natural injection. Let A = ( u ( A ) ,(A41 ~ E J ) be ) an algebra of type A ;the Z-reduct A1Z of A is defined as ( u ( A ) , (,fiAirliEZ>)wherefiA" =A$, for ieI. Hence A(Z is obtained from A by forgetting the operations h4 such that not ~ E T * ( I )However, . it should be observed that, for p j = O , although the operation h4 may be forgotten, its unique value h f ( 0 ) still remains an element of u(AlZ)=u(A). If A is a class of algebras of type A , the Z-reduct All of A shall be the class of all I-reducts of algebras of A. Let A be an algebra of type A and let X be a subset of u ( A ) ; then the closure [XIA1'is contained in the closure [XI". If ( A J s E S ) is a family of algebras of type A , then n ( A , l Z IsES) is the Z-reduct of n ( A , l s ~ S ) In . particular, for any algebra A of type A and for any non-empty set X , the algebra Op(A(Z,X ) is the I-reduct of O p ( A , X ) and, therefore, the inclusion u(H(AIZ, X ) ) C u ( H ( A , X ) ) holds. If Y C X , Y#O, the monomorphism r t i r I xfrom H(A11, Y ) into H(A/Z,X ) is the restriction of the monomorphism r$' from H ( A , Y )

EQUATIONAL MAPS

137

into H ( A , X);due to the construction of H ( A , 0) and H(AIZ, 0), this carries over to the case Y=O. Let X be a non-empty set, let P be an algebra of type A , absolutely freely generated by X,and let P' be the subalgebra of P II,generated by X . Then P' is an algebra of type A', absolutely freely generated by X. For let B= ( u ( B ) , (f,BliEZ)) be a non-empty algebra of type A' and let cp be in U ( B ) ~ . Define an algebra A = ( u ( B ) ,( h f ( j ~ J )that ) h; =A" for j = z ( i ) , whereas h; is an arbitrary operation of arityp, if notjez*(Z). Then cp has an extension h in Hom(P, A ) , and h r u ( P ' ) belongs to Hom(P', B). - Now let A be an arbitrary algebra of type A . The elements of Hom(P, A ) as well as those of Hom(P', AJZ) are in one-to-one correspondence with the elements of u(A)X=u(AIZ)X.Hence one obtains a bijection from Hom(P, A ) onto Hom(P', A l l ) . If A is not empty, let eA and eAllbe the epimorphisms from P onto H ( A , X) and from P' onto H(AIZ, X).If A is a class of algebras of type A , let Q(A) and Q(A1Z) be the sets of equations determined by A in P and by AIZ in P'. One obtains LEMMA 6. (a) If A is not empty then eAlr= eA 1u(P'). (b) For every class A : Q(AlZ)=Q(A)n(u(P') x u(P')). (c) For every class A: E(A)IZsE(A/Z), and if A consists of non-empty algebras also Es(A)[ZcEs(AIZ). For a proof, let cp be in U ( A ) and ~ let h,, hk be the extensions of cp in Hom(P, A ) and Hom(P', AlZ). Since hk=hv ru(P1),one finds eA(r)(cp)= h,(r)=hk(r)=eAlr(r)(cp)forevery rEu(P'). This proves (a), and (b) follows since the correspondence between h , and h i is one-to-one. Finally, (c) is a consequence of (b). From now on, assume that card(X) >rank ( A ) , and let a fixed A-coordinate system (K, ( p j l j ~ J ) for ) X be given. Let A be a non-empty algebra of type A and define B=AII. Then Supp(e,(s))=Supp(e,(s)) for every s e u ( P ' ) . Moreover LEMMA7. (a) If s ~ u ( P ' )and P.EK, P*(n)ESupp(eA(s)),then S ~ , " = S ~ , ~ . (b) If u ( H ( A , X ) ) = u ( H ( B , X ) ) then, for every n such that P,eK, also u ( H ( A , n))= u ( H ( B , n)). For a proof of (a), observe that seu(P') gives r ~ ( s A ~ " ) = e A ( s ) = e g ( s ) = r , " ( ~ ' , ~This ) . implies s A , n=sB," since r," are the restrictions, to u ( H ( A , n)) and u(H(B, n)) respectively, of the one injection r:(A)=r:(B)from S(u(A), n) into S(u(A), X ) . For a proof of (b), assume u ( H ( A , X ) ) = u ( H ( B , X)); since B=A/Z, one has always u ( H ( B , n ) ) s u ( H ( A ,n)). It follows from Lemma2and, incasen=O, from thedefinitions that u ( H ( A , X))=u(H(B, X ) ) implies u ( H ( A , X;P*(n)))=u(H(B,X;P*(n))). Now if d € u ( H ( A ,n)) then

rt,

138

W. FELSCHER

r t ( d ) E u ( H ( A ,X ; p*(n))); hence there exists d ' a ( H ( B , p*(n))) such that r;(d) =rK,,,(d'). But then d"eu(H(B, n)) can be found such that d'=j,(d), whence r,"(dj=r:(d"). Since r,", r," both are restrictions of the injection r;('), one obtains d=d" and d c u ( H ( B , n)). Let P be again the algebra ( u ( P ) , ( h j l j € J ) ) . Then one can prove LEMMA 8. For a non-empty algebra A one has u ( H ( A , X))=u(H(AIZ, X ) ) if and only if, for everyj not in 7*(Z), there exists s j e u ( P ' ) such that the equation (s,, hj(Pj)) holds in A . If u ( H ( A , X))=u(H(AII,X ) ) , then the existence of the elements s j follows from the fact that both maps eA and e A l rare onto u ( H ( A , X ) ) and e A I I = e Aru(P'). Assume now that the sj exist. Since H(AI1, X ) contains the generators e:', X E X , of H ( A , X),it suffices t o show that u(N(Al1, X ) ) is closed with respect to the operations h r of H ( A , X ) . This is clear i f j ~ z * ( Z ) . I f j not in T * ( I ) and y ~u (If(A lZ,X))'j, then A:(?)= [hf: q ] . But since ( s j , h j ( P j ) ) holds in A , and since s j ~ u ( P ' ) .Hence hfeu(H(AIZ,p,)). Since u(H(AIZ, X)) was shown to be closed with respect to superposition, it follows that [hf : q l ~ u ( H ( A I 1X, ) ) . The following remark concerns a different approach to the fact that the algebras H ( A , Z ) are closed with respect to superposition. Let A be an algebra of type A ' , let g be in u ( H ( A , Y ) ) and q € u ( H ( A ,2)j'. Assume now that Lemma 8 is available; define A by J = 1 u { j } wherej not in Z,p i = Y. Extend A to an algebra C of type A by adding the operation g . Let Xcontain Y and 2 ; then r:gEu(H(A, X ) ) and A=CIZ shows that r ; g = e A ( s ) = e , ( s ) for some seu(P'). Since r f g = e c ( h j ( P j ) ) by definition of C, one finds that (s, hj(Pj)) holds in C. Hence u(H(A, X ) ) = u ( H ( C , X)) by Lemma 8 and, if an injection pz is available, also u ( H ( A , Z ) ) = u ( H ( C ,2))by Lemma 7. Since [g : r ]= h ~ p ( A ' Z ) ( yone l ) , obtains that [ g : y] belongs to u(H(C, 2))and, therefore, to u ( H ( A , Z)). It remains t o give a direct proof of the second part of Lemma 8, making no use of superpositions. If the sj are given, it will be sufficient to show that for every u e u ( P ) there exists u ' ~ u ( P ' such ) that eA(u>=eA(v').Let M be the set of all elements u ~ u ( Phaving ) this property; since X c M , it suffices to prove that M is closed in P. If r E M P J ,then, by definition of M , a sequence x ~ u ( P ' ) ' j can be found such that e , . y = e A . X . If hr corresponds to j in H ( A , X ) , this shows eA(hj(q))= h r (e,. y) =hr(e, .x)= eA(h,(x)). Therefore, j ez *( Z ) implies h.( ) E U ( P ' ) whence , h j ( y ) € M . If not j e z * ( Z ) , define I e X X xl by I tfl*(yj)=X*p,; t p * ( p j ) and /z t X - P * ( p j ) the identity. Let g and g 1 be the extensions of I in Hom(P, P ) and Hom(P', P'); then g1= g ru(P1)and

EQUATIONAL MAPS

139

Since with ( s j i h j ( P j ) ) a's0 ( g ( s j ) , S(hj(Pj))> holds in A , one obtains eA(hi (q))= eA( h j ( ~ = ) )eA( g (hj(pj))) = eA( g (sj)). Then s j e u ( P ' ) implies g(si)=g' (sj), g ( s i ) E u ( P 1 ) ,whence h , ( q ) ~ M .

gfhj
4. Equational and functorial maps

Let A ' = ( n J i E Z ) and A2=(mklkEK) be two types and assume, for the sake of convenience, that I n K = 0. Put J = I u K and define A = ( p i [j c J ) by p , = ni if i E Z , P k = mk if kE K ; A is called the mixed type determined by A', A'. Let X be a set such that card(X) = rank(A) and let a fixed A-coordinate system (K, ( p j l j ~ J ) for ) X be given. Let P = ( u ( P ) ,( h , l j ~ J > )be an algebra of type A, absolutely freely generated by X ; let P ' = ( u ( P ' ) , ( f i l i ~ Z ) ) and P' =(u(P'), (g,lkEK)) be the algebras of type A' and A', absolutely freely generated by X , which are subalgebras of the reducts PII and PlK respectively. These notions will be kept fixed throughout this and the following section. Let B be a class of algebras of type A' and let C be a class of algebras of type A'. A function @ from B into C is called a map if u ( B ) = u ( @ ( B ) )for every BEB.A map 4 is called an equivalence if it is a bijection from B onto C. A map 4 is called functorial if Hom(B, D)cHom(+(B), 4 ( D ) ) for all algebras B, D in B; an equivalence 4 is called functorial if 4 as well as 4-l are functorial maps. Thus a map 4 is functorial if it determines a functor from the full category of algebras, determined by B, into the full category of algebras, determined by C,and if this functor commutes with the underlying set functors. If the classes B and C are equational, then these functors are precisely the algebraic functors in Lawvere's [111and Linton's [121categorical presentation of universal algebra. From now on, let B, C be classes of non-empty algebras of type A', A' respectively. Let 4 be a map from B into C. If BEB, B= ( u ( B ) , (A'liEZ)), $(B)= C = ( u ( B ) , ( g " l k ~ K ) ) define , an algebra Y l ( B ) of type A by Y,(B)=A= ( u ( B ) , ( h f l j ~ J ) ) hf=fiB , if i E Z , h$ =gE if kEK. Let A be the class of all algebras Yl (B) for BEB; A is called the class of mixed algebras determined by 4. Then Yl is an equivalence from B onto A and, for any BEB, 4 ( B ) = Yl(B)lK. If, moreover, 4 is an equivalence from B onto C, one defines analogously an equivalence Y 2 from C onto the same class A such that 4= Y ; Yl. If the map 4 is functorial, then Y, is a functorial equivalence. Let # be a map from B into C and let A be the class of mixed algebras determined by 4. The map # is called equational if there exists a sequence ( s , l k E K ) such that s k ~ u ( P 1for ) k E K and, for all algebras AEA and all

140

W. FELSCHER

kEK, the equations (sk, hk(&)) hold in A . In that case, 4 is said to be defined by these equations. An equivalence 4 is called equational if 4 as well as 4-l are equational maps. Obviously, any map defined on a singular class B is equational. Every equational map determines a representation of C in (the category determined by) B defined by identities in the sense of Cohn [2], IV.4, and all examples discussed in Cohn [2] arise in that way. However, there are occasions when representations defined by identities in Cohn’s sense do not arise from equational maps: for instance, the representation of R-modules (where R is a commutative ring) in R-algebras such that the universal functor assigns to every module its exterior algebra. If 4 is an equational map from B into C, defined by the equations {(sk, hk(&))lkEK), then 4 is uniquely determined by these equations. For if BEB and C = 4 ( B ) , A = Y , ( B ) , then (Sk,hk(&)) holds in A , whence g C -h A -(h,(P,))A,mk=~$’mkfor every k E K . Since B = A I I and s ~ E u ( P ~ ) , = s : ’ ~ ‘ . Thus g: = si’m k for every kE K. Lemma 7 implies stTrnk The following remark answers a question posed by H. J. Hoehnke. If 4 is an equational map from a class Es(B) onto some C, then 4 is uniquely determined by its restriction 4 IB. For let 6 be another equational map from Es(B) into some such that 4 rB=$ rB; let {(sk, hk(&))IkEK} and {(&, hk(Bk))lkEK}be the defining equations o f 4 and respectively, and let Y,, P, be the equational equivalences onto the corresponding classes of mixed algebras. If BEB and A = Y,(B)=PI(B) then hk(Pk)), (&,A,‘(&)) both hold in A . Hence (sk, f k ) holds in A and, by Lemma 6, also in AlI=B. Since BEB was arbitrary, ( s k , f k ) belongs to Q(B) for every ~ E and, K therefore, holds in every DEEs(B). Now if DEEs(B) and G = Y,(D) then (sk, f k ) and ( s k , h k ( P k ) ) both hold ill G, whence ( 4, h k ( p k ) ) holds in G for every k e K . Thus 4 can be defined by the same equations as r$ and, therefore, coincides with Every equational map 4 from B into C is functorial. For a proof, it will be sufficient to show that the equivalence Ylfrom B onto A is functorial. If B, D are in B, then Hom(Yl (B), Y l ( D ) ) cHorn(& D)is obvious. Define A = Yl(B), G = Y,(D); let g be in Hom(B, D) and ~ E KSince . ( s k , hk(Pk)) holds both in A and G, one has S k A ’ m k = ( h k ~ k ) ) A ’ m k = h k A , sFfmk= h;. But it was shown in section 2 that, for every $ E U ( A ) ~one ~ , has g(s$.””($))= s,“. mk (g .$4). Conversely, there are examples of functorial equivalences such that 4 is equational but not 4. Namely, let G be the class of all groups, written additively and viewed as algebras C= (u(C), ( 0 , -,+)) of type ( 0 , 1,2); let M be the class of all reducts of groups, obtained by omitting the unary

c

6

($9

6.

EQUATIONAL MAPS

141

operation - ; thus the elements of M are certain monoids. Let 4 be the map which assigns to every BEM the uniquely determined group whose reduct is B. Obviously, (b is a functorial equivalence and 4-l is equational. According to rank(d)=w, let X be countable and let P ' = ( u ( P ' ) , (O,+)) and P = P 2 = ( u ( P ) ,( 0 , - ,+ )) be the corresponding algebras, absolutely freely generated by X . Assume now that B is a subclass of M such that 4 1B is equational onto a subclass C of G. This will occur if and only if there exists s ~ u ( P ' )such that (s, -xo) holds in every group CEC or, equivalently, holds in the group F = P / Q ( C ) . Since P*(l)ESuPp(eF(-xo)), this is equivalent to =sF*', i.e. -eEpl =sFjl in H(F, 1). Defining M=$-'(F), Lemma 6 (a) shows that the elements d€u(H(F,X ) ) such that d=e,(s) for s ~ u ( P ' )are precisely the elements of u ( H ( M , X ) ) . Hence the elements of u(H(F, l)), representable in the form sF,' for s ~ u ( P ' ) are , precisely the elements of u ( H ( M , 1)). Therefore, 4 t B is equational if and only if -e;'Eu(H(M, 1)). Since H ( M , 1) is the monoid, generated by e2' in Op(M, l), one obtains that this is the case if and only if there exists n, O
LEMMA 9. Let g be the epimorphism from P onto D which extends the identity on X . Then, for every A E A and every r E u ( P ) , the equation ( r , g (Y)) holds in A . For let A be in A and let M be the set of all r ~ u ( P such ) that eA(r)= e A ( g ( r ) ) ;since X G M is obvious, it remains to show that M is closed in P. If q € M p j , then e , . q = e A . g q by definition of M . If h r is the operation in H ( A , X),corresponding to hj, thisimpliese,(hj(q))=hr(e,.q)=hr(e,.gq)= e A ( h j ( g q ) ) .Therefore it suffices to prove that eA(hj(gq))=eA(g(hj(q))).IfjEZ then h j u ( P ' ) =&=hq, whence hj(gq)=hq(gq) = g (Aj(?)). Assume now j e K .

142

W. FELSCHER

Since 6,;Bj=gq and since t(6,,) is an endomorphism of P, one has (hj ( B j ) ) = hj(t(dg,). P j ) =hj(ag,* P j ) =hj(gq)* Since with (S j , j(Pj)> also (t(d,,) (sj), t(dg,) (hj(Pj))> holds in A , one obtains eA(hj(gq))= eA(t(sgJ (hj(Pj))> = e A (t(8gJ ( s j ) )= e A (hp(gq)) = e A (9(hj(~1)). Let 4 be a functorial map from B into C (the case that B and C contain the empty algebra - if it exists - is admissible here). If B, D are in B and B is a subalgebra of D, then 4 ( B ) is a subalgebra of 4 ( D ) ; if B is a homomorphic (isomorphic) image of D, then 4(B) is a homomorphic (isomorphic) image of +(D).Let ( B J s E S )be a family of algebras in B and assume that the product B = n ( B s l s ~ S )belongs to B. Then Cp(B)= ( 4 ( B J l S E S ) since 4 4 (B))=u(B)= (u(Bs)l=S) = ( 4 4 (Bs))lSE S > and every projection from u ( 4 ( B ) ) onto u(4(BS))is homomorphic. In particular, the functor determined by (p preserves products and equalizers. Therefore, if B is closed with respect to products and subalgebras, then the general adjoint functor theorem (cf. e.g. Lawvere [I I], Th. 1.4, also Cohn [2], Th. 111.4.2 or Felscher [7] 2.2.4) ensures that this functor has an adjoint: there exists a function 6 from C into B and, for every CEC, a homomorphism fc from C into 4 ( 6 ( C ) )with the following universal property: for every BEB and every hEHom(C,4(B)) there exists a unique gEHorn(B(C), B ) such that h = gfc, (For algebraic functors in the sense of Lawvere this is the Theorem in IV, sect. 2 of [I 11; a generalization to a wider class of functors is prop. 5 in Linton [12]. For equational maps 4 the existence of 0 follows also from Cor. IV.4.2 in Cohn [2].) Let 4 be a functorial map from B into C, let A be the class of mixed algebras determined by 4 and let Y l be the functorial equivalence from B onto A. If an algebra B is B-freely generated by a set Y, then Y,(B) is A-freely generated by Y. For put A = Y , ( B ) and let G be in A; then u(G)'= u(GJZ)', G = Y , (GJZ) by definition of A and Hom(A, G)=Hom(B, G J I )by functoriality. Hence every 'p~u(G)'has a unique extension in Hom(A, C). Further, B=AII implies [ Y I B s[YIA;therefore Y generates A . - Even if 4 is a functorial equivalence, Y will in general not also generate 4 ( B ) , hence 4 ( B ) need not be C-freeIy generated by Y. However, if 4 is a functorial equivalence from B onto C and if C contains an algebra C, C-freely generated by a set 2 equipotent with Y, then the same reasoning shows that Y,(C) is A-freely generated by Z . Thus Y l ( B ) and Y , ( C ) are isomorphic and, consequently, so are (p(B)and C. In that case therefore (p(B)is C-freely generated by Y. Let 4 be a functorial map from B into C and let Y , be the functorial equivalence from B onto the class A of mixed algebras determined by 4. t(dgq)

n

n

JJ

EQUATIONAL MAPS

143

Let R be an algebra of type A , absolutely freely generated by a set Y , and let R' be the subalgebra of R(I, generated by Y. Let Q(B) and Q(A) be the equations of R' and R, holding in B and A respectively. LEMMA10. Let B contain an algebra B, B-freely generated by a set Z equipotent with Y. Then for any reu(R) there exists s € u ( R 1 )such that ( r , s)EQ(A). For a proof, let nl and n be the natural epimorphisms from R 1and R onto R1/Q(B) and R/Q(A) respectively. Since card( Y )= c a r d ( 2 ) there exists a bijection 6 from Y onto 2 which can be extended to gEHom(R, Yl(B)). Since Yl(B)is A-freely generated by 2, g factors into g=hn where h is an isomorphism from R/Q(A) onto Y , (B). Further, h is also an isomorphism from (R/Q(A))II onto Y,(B)II=B. Since R' is a subalgebra of RII, g l = g ru(R') is the unique extension of 6 in Hom(R', B ) and decomposes into gl=ll,n, where 11, is an isomorphism from R'/Q(B) onto B. Therefore f = h - ' . h l is an isomorphism from R'/Q(B) onto (R/Q(A))lI. Since g(s)= g1 (s), hn(s)=hlnl (s) for any s ~ u ( R ' ) ,one also has fnl (s)=n(s). Being an isomorphism, f is in particular onto u((R/Q(A))II) = u(R/Q(A)). Consequently, for every r ~ u ( R )there exists s ~ u ( R ' )such that f n l ( s ) = n ( r ) . Since also fnl(s)=n(s), this gives n(r)=n(s), ( r , s)€Q(A). - As an immediate consequence, one obtains

THEOREM 1. Let B contain an algebra, B-freely generated by a set of cardinality raizk(d). Then every functorial map from B into C is equational. If, moreover, C contains an algebra, C-freely generated by a set of cardinality rank(d), then every functorial equivalence from B onto C is equational. For equational equivalences between quasi-primitive classes of finitary algebras, this is Theorem 6 of Malcev [13]. It follows from earlier remarks that the assumptions with regard to equivalences from B onto C are satisfied if B contains an algebra, B-freely generated by a set of cardinality rank(A), and if C is closed with respect to subalgebras. Let I#I be a map from B into C and let Ylbe the equivalence from B onto the class A of mixed algebras determined by 4. If 4 is equational then Lemma 8 implies u ( H ( d ( B ) ,X ) ) c u ( H ( Y 1 ( B )X, ) ) = u ( H ( B , X ) ) for every BEB. Let 4 be arbitrary and assume, conversely, that u(H($(B), X ) ) G u(H(B, X ) ) for every BEB. Then also u(H(Y,(B), X ) ) = u ( H ( B , X ) ) holds for every BEB. Namely, let B be in B and abbreviate Yl( B )= A , 4 ( B )= C ; let k be in K. Since u ( H ( C , X ) ) G u ( H ( B ,X ) ) and e c ( h k ( / ? k ) ) E U ( H ( C , X)), there exists S k E u ( P 1 ) such that e,(Sk)=ec(hk(/?k)). Since eB=eA ru(P') and e C = e A lU(P2),this gives e A ( S k ) = e A ( h k ( P k ) ) , i.e. ( s k , h k ( / ? k ) ) holds in A .

144

W. FELSCHER

Now Lemma 8 gives u ( H ( A , X ) ) = u ( H ( B , X ) ) . - It would be interesting to have criteria which, if u ( H ( 4 (B),X ) ) _c u(H(B, X ) ) for every BEB, ensure that 4 is equational. A sufficient condition is that the class A contains an algebra A , functionally free for A. For then u(H(AlK, X ) ) E ~ ( H ( AX ,) ) = u(H(A(1, X ) ) shows that the right equations hold in A and, therefore, in A.

5. Construction of equational maps The conventions, agreed upon at the beginning of Section 4, shall be kept in effect throughout Section 5. Let (s,lkEK) be a sequence of elements of u(P'). A non-empty algebra B of type A' is called admissible for ( s , l k e K ) if B*(mk)ESupp(es(s,))for every k e K ; a class B of algebras is called admissible if every of its elements is admissible. If B=(u(B), ( f i " l i ~ 1 ) )is admissible for ( s , l k ~ K ) , an algebra Y,(B)=A=(u(B), ( h f [ j E J ) ) of type A may be defined by h f = fi" if i E I , h$=skSsmkif kEK. If B is a class of non-empty algebras of type A', admissible for (s,lkEK), then the class A of all algebras Y l ( B )for BEB will be called the class constructed from B and the sequence ( s , ( k e K ) . From now on, let B, C be classes of non-empty algebras of type A', A 2 respectively.

11. (a) If B is a non-empty algebra of type A', admissible for LEMMA (s,lkeK), then, for every k e K , (s, hk(jjk)) holds in Y,(B).(b) If G is a non-empty algebra of type A and if, for every kEK, (s, hk(Pk))holds in G, then G(Zis admissible for ( s , ( k e K ) and G = Y , (GIZ). For let B be given in (a) and define A = Y , (B). Since A ( I = B , Lemma 7 implies = s!,~*, whence s$mk=hi = (hk(Bk))ATmk. Thus (s, hk(Bk)) holds in A . Now let G be given in (b) and define B= G ( I . Since es(sk)= e, (s,) = eG(h, (&)), one has p* (m,) E Supp (es(sk));hence B is admissible and A= Y,(B)can be defined. Since the equations (s, hk(Pk)) hold both in G and A , one has sF'mk=hFand s t * m k = h fSince . s,Eu(P') and B = A I I = G ( I , Lemma 7 implies sz3*' =~ f "="s$"~ ", whence h,"= hf for every k E K . This proves G = A . For the following considerations it will be useful to remember that rank(A1)< rank(d)and rank(A2)< rank(A).Hence, if a class B is equational, the defining set M of equations has t o be taken from an algebra P'' of type A', absolutely freely generated by a set X, such that card(X,)=rank(A'). Therefore, X, may be chosen such that X,_cX and the algebra P'' may be taken to be the subalgebra of P', generated by X,.Obviously, B then is also defined by the set M of A'-X-PI-equations. Conversely, if a set N of

~t~~~~~

EQUATIONAL MAPS

145

A'-X-P'-equations is given, the class B defined by N is primitive and, therefore, definable by a set M of A'-X,-P"-equations. However, if rank(d') < rank(A) then no uniform construction of M from N is available. THEOREM 2. Let B be admissible for (sklkeK), let A be the class constructed from B and ( s k l k e K ) and let Y l be the map from B onto A. Then (a) Y1 is an equational equivalence from B onto A, defined by { ( s k , h k ( P k ) ) I k ~ K(b) } . If B is strictly equational and defined by a set M of A'-X-P'-equations, then A is strictly equational and defined by M u { ( s k , h k ( l j k ) ) l k E K } . (c) The strict equational closure Es(B) of B is admissible for ( s k l k e K ) , and the strict equational closure Es(A) is the class constructed from Es(B) and ( s k l k e K ) ,i.e. Es(AII)=Es(A)IZ. Obviously, (a) is an immediate consequence of Lemma 11. Let B be given in (b), let B be in B and define A = Y l ( B ) . Since e,=e,,,=e, ru(P'), the equations from M also hold in A . On the other hand, let G be a nonempty algebra of type A in which the equations from M as well as all ( s k ,hk(ljk)), k e K , hold. Then the equations from M also hold in GIZ, whence GIZEB. Consequently, Yl(GIZ)EA, but Y l (GlZ)= G by Lemma 11. For a proof of (c), let F be the algebra P'/Q(B). Since Q(F)=Q(B)= Q (Es(B)), Lemma 4 implies p* (mk)E Supp (eF(sk)) and, therefore, P*(mk)~Supp(e,(s,))for every DEEs(B) and every kEK. Hence the equations defining Yl also define an equational equivalence 9, from Es(B) onto the class K of all non-empty algebras of type A defined by Q(B)u{(s,,h,(P,))lk~K}. Since coincides with Yl on B, one has A c K and, therefore, Es(A)cK. It now will be sufficient to show that Q(A)=Q(K). First, ASK implies Q(K)cQ(A). In order toprove the other inclusion, assume (0, w)EQ(A). Since Lemma 9 can be applied to theequational equivalence PI, there exist u', w 1 in u ( P ' ) such that ( u ' , v)eQ(K), (w', w)EQ(K). Here Q(K)sQ(A) implies ( u ' , ~ ) E Q ( A ) ( , d ,w)eQ(A). Since Q(A) is a transitive relation, it follows that ( d , w')EQ(A). Since B=AIZ, Lemma 6 gives ( u ' , W'>EQ(B). Hence Q(B)zQ(K) implies ( u ' , w')eQ(K). Since also Q(K) is transitive, one obtains (0, w)eQ(K). THEOREM 3. If 4 is an equational map from B into C, then 4 can be extended to an equational map $ from Es(B) into Es(C), defined by the same equations as 4. If 4 is an equational equivalence from B onto C, then $ is an equational equivalence from Es(B) onto Es(C). For let A be the class of mixed algebras determined by 4 and let 4 be defined by {(sk, hk(Pk))IkEK) where S k E u ( P 1 ) . Since these equations hold in every AEA and since B=AIZ, it follows from Lemma 11 (b) that B is

146

W. FELSCHER

admissible for (sklkEK) and that A is the class constructed from B and (s,lkEK). Moreover, the equational equivalence Y1 from B onto A determined by 4 is the same as the equivalence determined by (sklk€K) according to Theorem 2 (a), since these equivalences are defined by the same equations. Thus Y l can be extended to an equational equivalence from Es(B) onto Es(A). Since A I K c C implies Es(AIK)sEs(C) and since Es(A)IKc Es(A1K) by Lemma 6, may be defined by $ ( B ) = p l ( B ) / Kfor BEEs(B). If 4 is an equational equivalence from B onto C and Y , is the equational equivalence from C onto A determined by 4-', then Y , can be extended to an equational equivalence p, from Es(C) onto Es(A), and one obtains = Fi pl.- An immediate consequence is

6

6

'.

COROLLARY 1. Let B and C be equationally equivalent classes of algebras. If B is strictly equational then so is C. It follows from Theorem 3 and Theorem 1 that it does not depend on the chosen A-coordinate system whether a map 4 from B into C is equational or not. For if 4 is equational with respect to a certain A-coordinate system, it may be extended to an equational map 6 from Es(B) into Es(C). Now 6 is functorial and Es(B) contains algebras, Es (B)-freely generated by arbitrary sets. Hence 6 is equational also with respect to every other A-coordinate system, and so is 4. - Further, Theorem 3 may be used in order to obtain defining equations for an equational map in a more economical way : COROLLARY 2. If 4 is an equational map from B into C then 4 can be defined by equations {(sk, hk(Pk))lkEK} such that, for every kEK, the element sk belongs t o the subalgebra P; of P', generated by P*(mk). By Theorem 3 it will be sufficient to prove this in case B is strictly equational; further, it can be assumed that B is not singular. Now let k be in K and define Y=p*(m,); let R and R' be the subalgebras of P and P' respectively, generated by Y, whence R ' = P i . Since Pk(/Q(B, Y ) is 3-freely generated by a set equipotent with Y, the assumptions of Lemma 10 are satisfied for the equational, and therefore functorial, map 4 . Since h k ( / ? k ) E U ( R ) , an element s k E u ( R 1 ) can be found such that (sk, h k ( & ) ) belongs to Q(A, Y)cQ(A), where A is the class of mixed algebras determined by 4. - Another application of Theorem 3 is COROLLARY 3. Let 4 be an equational equivalence from B onto C. Then (a) if B E B and B is generated by a non-empty set Y then 4(B)is generated by Y; (b) if B E B and B is B-freely generated by a non-empty set Y then

EQUATIONAL MAPS

147

& ( B )is C-freely generated by Y ; (c) if B E B and B is functionally free for B then q5(B) is functionally free for C. Observe first that an algebra, B-freely generated by Y, is also Es(B)freely generated by Y ; likewise, an algebra functionally free for B is also functionally free for Es(B). Therefore one may assume that B and C are strictly equational. Now let B be given in (a) and let D be the subalgebra of & ( B ) ,generated by Y . Since C=Es(C) is closed with respect to non-empty subalgebras, D belongs to C. Hence &-'(D) is a subalgebra of By containing Y, which implies q5-' ( D )= B, D = 4 (B). Further, (b) follows from (a) and the fact that q5 is functorial. Finally, let B be given in (c). Then B consists of the non-empty algebras in HSPIS({B}). Since C is primitive, the nonempty algebras in HSPIS({$(B)}) form a subclass of C. Since q5 is also functorial, q5 maps B onto this subclass of C. On the other hand, q5 maps B onto C. Hence C consists of the non-empty algebras in HSPIS({$(B)}), i.e. d ( B ) is functionally free for C. Let (sklkEK) be a sequence of elements of u ( P ' ) , let B be admissible for (SkIkEK), let A be the class constructed from B and (SkIkEK) and let Y l be the equational equivalence from B onto A. For B E B define q5(B)= Y, (B)IK, and let C be the class of all algebras q5(B) for BEB. The sequence (s,lkEK) is called complete with respect to B if q5 is an equational equivalence from B onto C. By Theorem 3 completeness with respect to B entails completeness with respect to Es(B). 12. Let B contain an algebra B, functionally free for B. Then LEMMA ( s , l k ~ K )is complete with respect to B if and only if u(H(B, X ) ) = u(H(&(B), Here it is clear that completeness of (sklkEK) implies already u(H(D,X ) ) = u(H(+(D),X ) ) for every D E B . Assume now that B E B is functionally free for B and that u(H(B, X>)=u(H($(B),X ) ) . Since Y , is equational, also A = Y,(B)is functionally free for A and u(H(B,X ) ) = u ( H ( A , X ) ) holds; hence u ( H ( A , X))=u(H(AlK, X)). Now Lemma 8 gives the existence of a sequence ( t i l i E Z ) in u(P') such that, for every iEZ, the equation (hi(Pi),t i ) holds in A and, therefore, in every GEA. Hence P*(ni)ESupp(eG(ti)) for every GEA and every ~ E Z ;since e G ( t , ) = e G I K ( t one i ) obtains that C is admissible for ( t J i E Z ) . Let Y , be the equational equivalence from C onto the class K constructed from C and ( t i \ i E l ) . Since the equations ((hi(Pi),t , ) l i € Z } hold in every GEA, Lemma 11 (b) gives G = Y , ( G l K ) for every GEA, i.e. A = K . Therefore 4 = Y, * Y, is a bijection and, moreover, an equational equivalence.

m.

148

W. FELSCHER

A rather peculiar criterion for completeness is given by

. ( s k l k e K )is complete LEMMA 13. Let B be admissible for ( s , ( k ~ K )Then with respect to B if there exists a function ( i ( k ) l k E K ) from K onto l a n d if, for every kEK, there exists an automorphism gk of P' such that g k ( f i ( k ) (Pi(k)))=Sk.

Observe first that gk YX is a bijection of X onto X,because P' is absolutely freely generated by X . Hence (gk X)-' induces automorphisms p k , p : , p t of P, P I , P 2 respectively such that p : = p k ru(Pi),p : = p k r u ( P 2 ) and p : =g;l. Since h k ( P k ) ~ u ( P 2it) , follows that the element ~ ~ ( ~ ) = p : ( h ~ ( P ~ ) ) holds. Since (sk, hk(Pk)) holds in lies in u ( P 2 ) . Further every AEA, also (fi(k)(Pi(k)), titk,) holds in every AEA for every k e K . Now let ( k ( i ) l i E l ) be a function from I i n t o K such that i(k(i))=ifor every ieZ (the axiom of choice may have to be used here). Defining t i = t i ( k ( i ) )one , obtains a sequence ( t i l i e l ) of elements of u ( P 2 ) such that ( f i ( P i ) , t i ) holds in every A E A for every i E I . Now the same reasoning as in the proof of Lemma 12 can be applied. Let ( s , l k ~ K )be a sequence of elements of u ( P ' ) ; let B be the class of all algebras of type A', which are admissible for ( s , l k e K ) ; let Y , be the equational equivalence from B onto the class A constructed from B and ( s , l k e K ) . A function g from u ( P ) into u ( P ' ) is called reductivefor ( s , l k ~ K ) if, for every r e u ( P ) , the equation ( r , g ( r ) ) belongs to Q(A). It follows from Lemma 9 that reductive functions always exist. In case B is the class of all non-empty algebras of type A', the algebra D, considered in Lemma 9, simply becomes Y , (P'), and the proof then can be simplified considerably. THEOREM 4. Let 4 be an equational equivalence from B onto C,given by equations { ( s k , hk(Pk))lkeK}for 4 and { ( h i ( P i ) , t i ) l i E I } for 4-l. Let g be a function from u ( P ) into u(P'), reductive for ( t i l i e l ) . Let B be strictly equational and defined by a set A4 of A'-X-P'-equations. Let g * ( M ) be the set of all d2-X-P2-equations (g(u), g(v)) for (u, u ) E M . Then C consists precisely of the non-empty algebras C of type A' such that (i) C is admissible for ( t i l i ~ Z ) , (ii) the equations from g * ( M ) hold in C , (iii) the equations {(g(sk), h k ( P k ) ) l k ~ Khold } in C . Since due t o Lemma 3 also property (i) can be expressed with help of equations, one obtains in this way a set of defining equations for C. For a proof, let A be the class of mixed algebras determined by 4. Since A is also the class constructed from B and (s,lkeK), A is strictly equational and defined by M u { ( s k , hk(&))lkeK}. On the other hand, A is the class

EQUATIONAL MAPS

149

constructed from C and (t,\iEZ), whence (g(sk), sk) for kEKand (g(u), u), ( g ( v ) , u ) for (u, U ) E Mbelong to Q(A). By transitivity then the equations in (ii), (iii) belong to Q(A) and, since C=AlK, to Q(C). Conversely, let C be a non-empty algebra of type A 2 with properties (i), (ii), (iii). Since (i) holds, Y , ( C ) can be defined; since C = Y,(C)IK, it will be sufficient to show that Y , ( C ) E A . As g is reductive for (tJiEZ), in Y , ( C ) the equations (g(sk),sk) for kEK and (g(u), u), (g(u), v) for (u, V ) E Mhold. Since the equations holding in C also hold in Y 2 ( C ) ,one obtains that the equations from M u { ( s k , hk(Pk))IkEM}hold in Y , ( C ) . In concluding this paragraph, a theorem will be formulated for which the type A' shall begiven, while the type A' is to be determined in a particular way: THEOREM 5. Let B be a class of non-empty algebras of type A'. Then an ordinal type A 2 and a class C of non-empty algebras of type A' can be found such that (i) B and C are equationally equivalent; (ii) for every kEK: the ordinal number mk is a cardinal number; (iii) there exists a bijection (k(i)li€Z) of I onto K such that, for every i E I , mk(j) is the smallest cardinal number w ifor which a set Y,EX exists such that card(Yi)=wiand,foreveryBEB, YiESupp(eB(fi(Pi))). Moreover, if the type A' is ordinal, then the injections jk,kEK, of the A-coordinate system may be chosen such that pi( j ) =pi 1mk(j ) for every i e I. For a proof, let K be such that ZnK=O and let there exist a bijection (k(i)liEZ) from Z onto K. Since A' is given, there exist fixed injections p i from n, into X where card(X)=rank(Al). Since p*(ni)€Supp(e,(f.(pi))) for every BEB, sets Yimay be chosen such that Yicj3*(ni) and card(Yi)= mk(,), where mk(i)is determined in (iii). In order to treat the general case, define & j ) as an arbitrary bijection from mk(i) onto Y, and define sk(i)= f i ( p i ) . Then B becomes admissible for (sk(illiEZ), and if A is the class constructed from B and ( $ k ( j ) I i E Z ) then (hi(fli),hk(i)(&(j)))holds in every A E A . Hence (s,,,)JiEZ) is complete with respect to B. Turning to the case that d l is an ordinal type, let the sets Yi again be chosen such that Y,Ej?*(ni) and card( Yi)= mk( i). Since mk( i) d IE j , one can define pk(,)= p i r Y ) ? k ( j ) . If mk(j)= O let g i be the identical automorphism of P'. Assume now that mk(j)>O. Let p i be a bijection from ( p i ' ) * ( Y i ) onto mk(,.);since both these sets are contained in ni, p i may be extended to a bijection hi of n, onto itself. Then the bijection piS,pLr of p* (n,) onto itself can be extended to a bijection y i of X onto itself which maps Yi onto P*(mk(,,). Now let g i be the automorphism of P' induced by yi, and define

150

W.FELSCHER

~ ~ ( ~ ) = g , ( f , (ItP now ~ ) ) .suffices to show that B is admissible for (&(i)IiEI), since Lemma 13 then will ensure that ( s k ( , ) 1 i ~ is I )complete with respect to €3. Let B be in B. Since P*(mk(i))ESupp(eB(sk(i,)) is clear if mk(i)= O or if B is singular, assume that rn,(,,>O and let B be not singular. Then H (B, X ) is E((B})-freely generated by (e;'(xEX}; since H(B, X ) itself belongs to E ( ( B } ) , yi determines an automorphism g f of H ( B , X ) such that g F ( e y ) = e:,& for X E X . Then e,.g,=g"e, since these homomorphisms coincide on X.Now g? ( e B ( f i ( P i ) ) ) = e B ( g i ( f i ( P i ) ) ) = e B ( s , ( i ) > ; hence yiESuPp(eB(fi(Pi))), i.e. e,(f,(P,))Eu(H(B,X ; Yi)), implies that eB(Sk(i)) belongs to the image of u(H(B, X ; Y,)) under g y . Since Y,#O, H ( B , X;Yi) is generated by (e,BXlxEY,}; hence g f maps H ( B , X;Yi) onto H(B, X ; y: ( Yi))= H ( B , P*(mk(i))).Thus e B ( s k ( i ) ) E U ( H ( B , X;P*(rnk(i)))),p*(mk(i))E Supp (eB (sk( i)).

x;

6. Definable maps and syntactical equivalences

A relational type shall be an ordinal type A = ( n, li€I ) such that 0
EQUATIONAL MAPS

151

For every set E, let R ( E ) = ( u ( R ( E ) ) ,(r:lleL)) be an algebra of type A L such that u ( R ( E ) ) = % ( E X )and the operations are defined as follows: if ZE(L, u L1), r: is the obvious Boolean operation (i.e. complementation, subjunction or intersection); if 1=(2, Y ) , r: is the Y-cocylindr$cation t,: if S s E X and cpeEX,then cpEtyS if and only if, for every XEE', cp 1- Y = x t - Y implies XES.Further, for every SGE' one defines S u p p ( S ) c % X by Y ~ S u p p ( 5 'if) and only if t-,S=S, i.e. if, for any cp, x in EX,cp 1 Y=x 1 Y implies that c p ~ Sis equivalent t o XES.Then Supp(S) again is a filter, because tytz= tYuZholds for all Y and 2. From now on, a fixed bijection P from 6 onto X shall be given. For every Y c X let n y and y y be uniquely determined by the properties ny<6, y y bijective from n y onto Y, and P - ' . y y strictly monotonic from ny onto (P-')*(Y) - in particular, if n<6 and Y=p*(n) then ny=n, yY=pIn. If E is a set, SGE', y ~ S u p p ( S )define , S(') in En' by $ES("if and only if there exists cp ES such that $ = cp * yy. S(') is called the relation of arity n y determined by S. S can be computed from S(') since cpcS if and only if cp * y y E S'Y'. Let Ars be the set of all fi(PTn,) in At. There exists a bijective correspondence between the class of all models A of type d and the class of all functions 7c from Ats into sets u ( R ( E ) )such that j?*(ni)~Supp(7c(fi(b 177,))). For if A=(u(A), ( f i A l i ~ Z > )is given, define n: as a function into u(R(u(A))) by c p ~ n . ( f ~ ( P r nif~and ) ) only if c p - P 1 n i ~ J Aif; n: is given, define A by u ( A ) = E and x.A=n:(fl(prn,))('?*(ni)). Now let A be a model and let 7cA be the corresponding function into u(R(u(A))). Extend nA to a function n: from Ar into u(R(u(A)))by setting c p ~ n : ( f ~ ( Aif ) ) and only if there exists xerc,(f,(P [n,))such that c p * A = x - P Tni (i.e. if and only if c p . A ~ f i ~ Let ) . eA be the extension of 7c: in Hom(P, R(u(A))).If rEu(P), cp~u(A)',then cp is said to satisfy r if cpee,(r). r holds in A if e A ( r ) = u ( A ) X ;this is abbreviated by lb r. If A is a class of models, then r holds in A - abbreviated by ki, r - if r holds in every A E A . If A is a model, then lkA (0-w) is equivalent to e A ( o ) = e A ( w ) ;for every formula r one has lb (V) r-r. Further, algebraic induction shows that fr(r)ESupp(eA(r))for every r e u ( P ) ; thus the filter Supp(e,(r)) has a base of sets of cardinality less than 6. In analogy to Lemma 3, the fact that Y ~ S u p p ( e , ( r ) )is equivalent to IFA r + + V Rwhere r R = f r ( r ) - Y ; if diin(d)= 6=0, then R is finite and YeSupp(e,(r)) is equivalent to IkAr-VJXr for every X E R . If A is a class of models, let Q ( A ) be the set of all sentences r such that lkA r, and let Qp(A) be the set of all open formulas r such that lb r. Let

152

W. FELSCHER

D(A) be the class of all models D such that ItDr for every rEQ(A), and let Dp(A) be the class of all models D such that IFD r for every rEQp(A). D(A) then is called the dejnable closure of A, and Dp(A) may be called the open closure of A. A class A is called dejinable if A=D(A); this is the case if and only if there exists a set M of sentences such that AEA is equivalent to it, r for all rEM. In this case, A is said to be dejned by M . The class A is called open if A=Dp(A); this is equivalent to the fact that A can be defined by a set of open formulas. The following theorem about the existence of substitutions for languages P will be proved in the appendix: For every function v] from X into X there exists a function sub,, from u ( P ) into u ( P ) such that (i) sub,, fi(A)=fi(v]*A) for fi(A)EAt, (ii) for every rEu(P), every model A and every q ~ u ( A ) ~ : qEe,(sub,r) ifandonly ifq*q~e,.,(r).It is a consequence of (ii) that, for every q and every A , FI, u u w implies It, sub,,v++subqw.Making use of substitutions, YeSupp(e,(r)) becomes equivalent to l;A r++subq(R)r,where q(R) maps R = f r ( r ) - Y injectively into the set of elements of X which do not occur in r, and y(R) is the identity outside of R . If dim(A)=d=w, let ( q ( x ) l x ~ Rbe ) a (finite) sequence of functions v](x) in X x such that q ( x ) maps x into an element of X , not occurring in r, and otherwise is the identity; then YeSupp(e,(r)) is equivalent t o IF, r+-+sub,(x)r for all ~ E R . Consider now, in analogy t o Section 4, relational types A'=(nJiEI), A2=(mk/k€K) withamixed typeA=(pj( j E J ) ; l e t 8 besuchthatdim(A)<8 and let p be a bijection from 6 onto a set X . Let P be a 8-A-language, defined with help of X and the identity (hjl j E J ) of J . Write hi=fi if iEI and h,=g, if keK, and let At', A t 2 be the sets of allfi(A) and g,(A) in At respectively. Let P ' , P 2 be the subalgebras of P, generated by At' and A t 2 . Let B be a class of models of type A' and let C be a class of models of type A2. A function 4 from B into C is a map if u(B)=u(+(B)) for every BEB;a map is an equivalence if it is a bijection from B onto C. If 4 is a map, one can define the class A of all models of type A determined by 4, and 4 decomposes into an equivalence Y , from B onto A, followed by the reduction map from A into C. A map 4 is called definable if there exists a sequence (sk(k€K) in u ( P ' ) such that, for every keK, IFA SkCthk(P rink);in that case, 4 is said to be defined by these formulas. If, moreover, the sk can be chosen to be open formulas, then 4 is called open. An equivalence 4 is called definable resp. open if both 4 and 4-l are so. Making use of substitutions, one sees easily that a map 4 is definable already if there exists a sequence (sklkEK) in u ( P ' ) and a sequence (MkIkEK) of sets M k E Xsuch that, for every model BEB, one has e,(sk)(Mk)=gEwhere 4 ( B ) = C = ( u ( C ) , (gtlkEK)). As in

EQUATIONAL MAPS

153

the case of algebras, a definable (or open) map is uniquely determined by its defining formulas, and a definable map 4 on a class D ( B ) (resp. an open map on a class D p ( B ) ) is uniquely determined by its restriction 4 1B. Let 4 be a map from B into C , definable by ( s , l k E K ) ; a statement analogous to Lemma 9 can be proved. Define a function g1 from At into u ( P ' ) as follows. If rEAt', put g l ( r ) = r ; if r = h k ( P r m k ) , put g l ( r ) = s k . If r=hk(IZ) and A€Xrnh,define q in X x by qlj3*(mk)=IZ.(j?lrnk)-' and q r X - p * ( m , ) the identity; then put gl(r)=sub,,sk. Let g be the extension of g1 in Hom(P, P'). Then IFA r - g ( r ) holds f o r every r E u ( P ) . For a proof, it is sufficient to show that, for every A E A , the homomorphisms e, and e,.g from P into R ( u ( A ) ) coincide on At. This is obviously the case for elements rEAt' or r=hk(P lm,). But if r=hk(A) then r=sub,(h,(P rm,)) with the q defined above; hence kI, r - g ( r ) since lb hk(P rmk)c)s, implies sub,(h,(p rm,))-sub,s,. - It should be noted that g not necessarily maps sentences into sentences again. However, if the formulas s, are open, then g preserves open formulas. The definition of models B, admissible for a sequence ( s , l k E K ) , can be taken verbally from Section 5. Further, Lemma 11 remains in effect together with its proof, if only phrases like "(s,, hk(Pk))holds in G" are replaced by Iti; Skc*hk(P 1m,). Likewise, Theorem 2 remains true if equational maps, defining equations and equational closures are replaced by definable maps, defining sentences and definable closures; moreover, if the sequence consists of open formulas, then open maps, defining open formulas and open closures can be considered. This translation is obvious for parts (a) and (b) of Theorem 2. As for (c), remember first that together with r also sub,,(,,r is open; hence if B is admissible and the sk are open, then also D p ( B ) is admissible. Consider now the case of definability and sentences. Let Ylbe the equivalence from B onto A and let PIbe its extension to an equivalence from D ( B ) onto K , where K is defined by Q ( B ) together with the sentences (V)s,c)h,(p tm,). Again, it has to be shown that Q ( A ) c Q ( K ) . Let g be a homomorphism from P into P' such that IkKr-g(r) for every r E u ( P ) . If r g Q ( A ) , then e,(gr)=e,(r)=u(A)' for every A E A . Hence the universal closure (V) gr belongs to Q ( A ) , and since g ( r ) E u ( P ' ) it belongs already to Q ( B ) . Then Q ( B ) s Q ( K ) implies (ti) g r E Q ( K ) . Thus one obtains for every K E K that u(K)'=e,((V) g r ) = e , ( g r ) = e , ( r ) , i.e. r € Q ( K ) . Since Theorem 2 is available now, also Theorem 3 together with Corollary 1 hold in the new situation. Further, Theorem 4 can be taken over without change. Although Lemma 13 and the algebraic machinery in the proof of Theorem 5 break down, Theorem 5 itself remains essentially in effect, and

154

W. FELSCHER

this even in the strong form that the equivalence from B onto C can be chosen as open. However, in order t o avoid the case m,(i,=O, it has to be assumed that for every i E 1 there exists at least one BEB such that not OESupp(eB(fi(PTn,))). Define then the functions p i , ai and y i as in the case of ordinal types of algebras. For every iEZ, define ~ , ( ~ ) = s u b ~TnJ= ~f~(/j fi(yi.p rni)=fi(P.6i). Then B is admissible for (sk(i)liEI), since Y E S u p p ( e B ( f i ( P mi))) imp1ies Y * ( Y i ) E S u p p ( e B ( s u b y , f i ( P mi))>, Y,*(&)= P * ( M k ( i ) ) . Since the s k ( i ) are open, there exists an open map 4 from B onto a class C of models of type A'; let A be the class of models of type A determined by 4. Since lb s k ( i ) ~ h k ( irm,(i,) ) ( p for every ~ E I , also IF. sub,, - 1 s k ( i ) H S U b y i- 1 h k ( i ) (p mk(i)), but Subyi- t S k ( i ) = f i ( p 1 n i ) . Define t i = m b y i - l h k ( i ) ( / j/ m k ( i , )for i E z ; then c becomes admissible for ( t i l i E z ) . Let Y 2 be the open equivalence from C onto the class K constructed from C and (tiliEZ). Since It;4fi(P / n i ) - t i for every AEA, the analogon of Lemma 11 gives again A = Y 2 ( A l K ) for every AEA, i.e. A = K . Thus ~ = Y , ' . Y , is a bijection and, therefore, an open equivalence. It is clear that definable maps and equivalences may as well be studied for relational systems which, in addition to relations, also carry a sequence of operations. However, aside from notational complications nothing new seems to arise in that case. For let ( A ; , A') be the pair, formed from the algebraic type A ; and the relational type A' of a class B of such relational systems. A map 4 from B into a class C with types (A;, A 2 ) will be definable if not only the relations belonging to A' are definable but also, for every A+operation p of arity n, the (n + 1)-ary relation p ( P rn)=P(n) becomes definable by A'-formulas. And in general, nothing more about the definability of A:-operations can be said. If, however, in a fortunate situation some A+-operations are definable already by A$terms, then the relationship between these operations and terms can be described completely with the algebraic techniques of Sections 4 and 5. For let Br be the class of algebras, underlying the relational systems in B, and let C T be the class of those algebras which are reducts, with respect to the algebraically definable operations, of the algebras underlying the relational systems in C. Then 4, being a map, establishes an equational map from BT into C T . The methods developed until so far can be used in order to deal with so-called syntactical transformations. Let A be a fixed relational type; let A L 1 , AL2 be types of algebras and let ALo be the mixed type determined by A L ' , AL2. Let 6 be an infinite regular cardinal number such that dim(A)<6, dim(dLo)<6, and let p be a bijection from 6 onto a set X . Define the set At as before; let P L Obe an algebra of type ALo, absolutely freely generated

r

EQUATIONAL MAPS

155

by A t , and let P L 1 ,P L 2be the subalgebras generated by At in the reducts of P L Owith respect to A L ' , A L 2 . Assume now that, for every set E, the set %Ex carries a well-defined algebra R' ( E ) of type AL1. Let 4 be an equational equivalence from the class of all algebras R' ( E ) onto a class of algebras of type A L 2 ; put 4 ( R ' ( E ) ) = R 2 ( E ) and Y l ( R ' ( E ) ) = R o ( E ) .Since the models of type A are in a bijective correspondence with certain functions 71 from Ats into the sets u(R1(E))= u(R2( E ) )= u(Ro(E)),every model A determines homomorphisms e i , e:, e: from PL', PL2, PLo into R'(u(A)), R2(u(A)), R o ( u ( A ) ) such that eiru(P")=e:, eiru(PL2)=e:. Let gl, g 2 be two reductive functions determined by 4, g1 from P L ointo PL' and g 2 from P L o into P L 2 ;gl, g 2 then are called syntactical transformations between PL' and PL2. If rEu(PLo) then the equations ( r , g l r ) , ( r , g 2 r ) hold in every algebra R o ( E ) ; hence ei(r)=e:(g'r), e i ( r ) = e i ( g 2 r ) for every A , i.e. lkA r-g'r, IFA r-g2r. Consequently for every r e u ( P L ' ) and every s c u ( P L 2 ) also IFA r-g1g2r and IFA s-g2g1s. Let r be in u ( P L o )and M s u ( P L 0 ) . For every model A , define M It-- r by ( e i (m)lmeM)G e i ( r ) . It follows immediately that M lb r implies g 2 * ( M )IFA g2r and g ' * ( M ) IFA glr. Define the closure operator Cs on %u(PLo)by r e C s ( M ) if, for every model A , M IFA r. Let Cs' and Cs2 be the closure operators on %u(PL') and %u(PL2), induced by Cs. Then, for every M c u ( P L ' ) and every N s u ( P L 2 ) , g2*Cs'(M)ECs2(g2*M), gl*Cs2(N)c cs' ( g I * N ) ,Cs' ( M )= cs' ((g1g2)* M),C?(N) = Cs"(gZgl)*N) hold. A certain inconvenience arises now from the fact that g2, say, will not always transform closed sets into closed ones. As a remedy, define for every s c u ( P L 2 )a unary operation h: on u ( P L 2 )by h:(v)=s if v=g2g1s, h:(v)=v otherwise. Obviously, sets closed with respect to Cs2 will be closed with respect to all operations h:. If M s u ( P L 1 )and Cs'(M)=M, let N be the set of all values under operations h: of elements in g 2 * ( M ) .Then N is closed with respect to Cs2 and, therefore, N = Cs2(g2*M). Let dim(Cs') be the smallest infinite regular cardinal number m such that, for every M and r, r E Cs' ( M ) implies the existence of a set M' such that M'E M , card(M')
n

156

W. FELSCHEK

Let F be an algebra of a type A F such that u(F)=u(PL'). F is called an axiomatization of Cs' at a set M c u ( P ~ ' )if C s l ( M ) = [MIF;F is called an axiomatization of Cs' if F is an axiomatization of Cs' at every set M . In this situation, the values of constant operations of F are called axioms, and the non-constant operations are called rules of derivation. It follows from well known facts about closure operators that there are always axiomatizations F of Cs' such that dim(AF)=dim(Csl). i t will be shown now that, given syntactical transformations gl, g2 and an axiomatization F of Cs' (an axiomatization F of Cs' at the empty set), there is an explicit method to transform F into an axiomatization G of Cs2 (an axiomatization G of Cs2 at ), a the empty set) such that dim(AF)=dim(dc). For every r ~ u ( P ~ 'define unary operation h,! on u(PL') by h,'(u)=r if u = g ' g 2 r , h,'(u)=u otherwise. If k u ( P L ' ) " for some ordinal number n, define a unary operation hi on u(PL1)nby h i ( $ ) ( m ) = h ~ ~ m l ( $ ( m for) ) every $eu(PL')n and every mO, and for every A€u(PL')ni, G shall have an operation&,: of arity ni, defined by J;.,:((x)= g x F ( h :*gl (Instead of the unary operations one might have introduced constant operations s+-+g2g1sfor every s e u ( P L 2 ) and added a binary operation, yielding the modus ponens.) The following observations will be useful. Assume that N G u ( P L 2 ) . Then g 2 maps M = [gl*NIFinto "1'. For if r = g'sand SEN, theng2g's~[NIG; further g2a€[NIGfor every value a of a constant operation of F. Assume now that $EM"' and that already g 2 . $ ~ ( [ N ] ' ) n L Then . fi:(g"$)~[N]', but fi: (g 2 $) = g x " ( h i -gl . g 2 $) = g2fiF($). - r f , in addition, [ g1*NIF= Csl(gl*N), then g' maps [N]' into M = [g1*NIF.Namely g2g1aECs1(M) for every set M ; further, if s€[N]' and g'sEM, then also g1g2g1sECs'(M) and g'h;(s)ECs'(M) for every t € u ( P L Z ) Finally, . assume that x ~ ( [ N l ' ) " ' and that already g l - X E M " ' . Then M = C s ' ( M ) implies h:.g'*xEM"' for every I , and M = [MIFimpliesfiF(hi.gl.x)EM. Hence M = C s l ( M ) implies g1g2fiF(h:*g1 . x ) E M ,but g l g x f ( h : * g l-x)=gx,?(X). In order to prove the assertions about G, it will be sufficient to show that, if F is an axiomatization of Cs' at g'*(N), then G is an axiomatization of Cs2 at N . Assume therefore [g1*NIF=Cs1(g1*N).Then g1 maps C s 2 ( N ) into Cs'(g'*N) and, by the earlier observation, g 2 maps [g1*NIFinto [NIG. Thus SECS'(N)implies g'g'sc [N]', whence SE [N]' since G contains the ex).

-

EQUATIONAL MAPS

157

operation h:. Therefore C s ' ( N ) c [N]' holds. On the other hand, by the other observation g1 maps [N]' into [g'*N]'=Cs'(g'*N), and g 2 maps Cs'(gl*N) into Cs2((g2g')*N).Thus SE [N]' implies g2g1sECs2((g2g1)*N). Since Cs2(N)=Cs2((g2g1)* N ) and since Cs2((N)is closed with respect to h i , one obtains SE Cs2( N ) , i.e. [N]' c Cs2( N ) . Syntactical transformations can be studied, in particular, with respect to the known (and finitary) axiomatizations of first-order logic. It is in this case that axiomatizations of Cs at the empty set become important, because here usually an axiomatization at the empty set is given first, and is only afterwards amended in order to obtain an axiomatization at every set. Also, it should be pointed out that the standard axiomatizations have an additional property in that they are formal. Namely, a rule f , yielding say modus ponens, may be conceived as constructed by the following process. Consider an algebra Q of type A L , absolutely freely generated by a set Y , and prescribe a sequence (yo,yo-+yl,yl) of elements of Q . Define f for r , s in u ( P L ) as follows: if there exists a homomorphism h from Q into P L such that h(y,)= r, h(y,-ty,)=s, then f ( r , s ) = h ( y l ) ; otherwise f ( r , s ) = r . Without pursuing this matter any further, it may be remarked that in the above transformation of F into G every formal rulefiF can be transformed into a formal rulefi' which comprehends all the rules fi:. Appendix

In this appendix, the existence of general substitutions will be proved. In generalization of the setting in Section 6 , languages with terms and models with operations will be admitted. So let A T be an ordinal type and let A be a relational type. A model A of type ( A T , A ) is an ordered pair ( a ( A ) , b ( A ) ) such that a ( A ) is an algebra of type A T , b ( A ) is a relational system of type A , and u ( a ( A ) ) = u ( b ( A ) ) ;this set is written as u(A). Let 6 be an infinite regular cardinal number such that dim(A7)<6 and d i m ( A ) < 6 ; let be a bijection from 6 onto a set X . Let T be an algebra of type A T , absolutely freely generated by X . If A = ( ~ J ~ E letI ()f ,i l i E I ) be the identity on I and define At as the set of all ordered pairs ( A , A)=A(A) where A€u(T)";. Let P = ( u ( P ) , (r,llEL)) be the algebra of type A L , absolutely freely generated by A t . On u ( P ) one defines recursively the ordinal-valued function deg such that deg(r)=O if rEAt and deg(r,(p))=sup(deg(p(m))+IlmEq,) if p ~ u ( P ) ~this ' ; can be made precise by introducing suitable operations on the set 6. There will be occasion to give proofs by induction and definitions by recursion on deg; a particular case is the induction on the number of

158

W. FELSCHER

quantifiers in usual first-order logic. In a similar recursive way, one defines functions part from u ( T ) into % u ( T )and from u ( P ) into % u ( T )u % u ( P ) : for t E u ( T ) , part(t) shall be the set of all subterms of t , and for rEu(P), part(r) shall be the set of all subformulas and subterms of subformulas of r. The only important convention here is that, for a formula V Y r , one has part(V,r)= Yupart(r). Finally, one defines the functionfr from u ( P ) into %X, assigning to every r the set of variables free in r. It follows from the choice of 6 that, for every formula r, card(part(r))
>

,

x

*I]

r-

159

EQUATIONAL MAPS

Y n p a r t ( q ( x ) ) = O , i.e. p a r t ( q ( x ) ) c - Y, whence h;q r f r ( r ) = h ; q r.fr(r) for every x. Since, moreover,fr( r ) E - Y ,one obtains $ rfr( r ) = h; q If.( r ) = h;qY t f r ( r ) = h ; q y f r ( r ) for every $ and every x. Thusevery $ determines a x such that x r - Y = q r - Y and h ; q y r f r ( u ) = $ r f r ( u ) : define x r Y = $ Y. Conversely, every x determines a b,t such that $ Y=h;q 1- Y and h ; q y fr(u)=$ r f r ( u ) : define $ 1 Y=x Y.

r-

r

(ii) Let q be such that, f o r some Y C X,f j = q 1 Y is a bijection on a set Z E X , while q - Y is the identity. Let r be a formula such that ZS -part(r). Then x E f r ( r e p , r ) /#(not X E Y and x E f r ( r ) ) or ( X E Zand f l - ' ( x ) E f r ( r ) ) . Proof. Since Z E - p a r t ( r ) implies Z c -part(w) for wEpart(r), induction in the set of subformulas of r can be applied. The statement is true for r E A t and carries through under the sentential operations. Assume now r = V, u. Then x E f r ( r e p , r ) i f f not X E W and xEfr(rep,,u). Here qw determines a bijection from Y - W onto 2 - q * ( Y n W ) ;hence induction gives xEfr(rep,,u) i f f(not X E Y - Wand x E f r ( u ) ) or ( x E Z - q * ( Y n W ) and Q - ' ( x ) ~ f r ( u ) ) . Since W ~ p a r t ( rimplies ) Z E - W, one obtains x g f r ( r e p , r ) iff (not X E Y and not X E Wand x E f r ( u ) ) or (xEZa nd not f j - ' ( x ) ~ Wand Q-'(x)Efr(u)) i f f (not X E Y and X E f r ( r ) ) or ( X E Zand f j - ' ( x ) ~ f r ( r ) ) . (iii) Let q, r satisfy the assumptions of (ii). Thenf o r every model A : e A ( V y r )= eA ( v Z r>. Proof. Assume V,uEpart(r) and X E fr(Vwu). Since W c p a r t ( r ) implies Z n W=O, it follows from not X E W that not ~ ( x )W. E Thus r, q are compatible. Now qEe,(V,r) i f ffor every x, x 1- Y = q Y implies xEe,(r); likewisecp Ee,(Vz rep,r)iff for every$, $ 1 -Z= -2implies $eeA(rep,r), i.e. h,.qEe,(r). Then h,-q 1- Y=$t-- Y and Z C - p a r t ( r ) shows that h @ - qr f r ( r ) - Y = q r f r ( r ) - Y. Thus every $ determines a such that x 1- Y=cp t - Y and x r f r ( r ) = h , * q r f r ( r ) : define x [ Y=h,.q 1 Y . Conversely, every x determines a $ such that $ 1- Z = q 1-2 and x r f r ( r ) = h , . ~r f t ( r ) : define $ rZ=x-fj-l / Z . Let q be in U ( T )and ~ let r be a formula. Definej(r, q ) to be the smallest ordinal number a<6 such that, for every a', a<'a'<6, one has (j) not P(r')Epart(r), and (jj) if x E f r ( r ) then not P(a')Epart(q(x)). Since 6 was chosen large enough, j ( r , q ) always exists. If Y c X , card(Y)<6, define q(r,Yfin X x such that q ( r , y 1) - Y is the identity, while ~ ( ~ , ~ ) ( y ) = / ? ( j ( r ,q ) + / ? - ' ( y ) ) for Y E Y. Obviously, q ( r , Y1) Y is a bijection onto a set Z E X such that 2s -part(r).

r

r-

160

W. FELSCHER

For every q, define by recursion on deg a function rut, from u ( P ) into u ( P ) : if rEAt then tut,r=r; let rut, be homomorphic with respect to the sentential operations; define tut,(V,r)=V,tut,,rep,,,,y,r, where Z is the image of Y under u ( ~ , , ) .Obviously, deg(r)=deg(tut,r) for every formula r. (iv) For every q, r and every model A:eA(tut,r)=eA(r). Proof. Apply induction on deg. The statement is trivial for rEAt and carries through under the sentential operations. Consider now V y r , whence tut,(Vyr) =V, tut,rep,,,,,, r. The choice of Z ensures that (iii) can be applied ; hence eA( V y r )= eA(Vzrep,cr,y,r). However eA(rePq(,.Y, r ) =eA(tutq rep,,,.,y, r ) by induction, which implies eA(V, r ) = eA(V, tut, rep,(,.y, r ) . (v) For every q, r: fr(tut,r)=fr(r). Proof. Apply induction on deg. The statement is trivial for rEAt and carries through under the sentential operations. Consider now V, r. Since Z c --part ( r ) , one obtains XE fr(V, rut, rep,,,r,Y)r) iff not XEZand XE fr(tut,repq(,,y,r) iff not XEZand X E fr(repqc,.y,r) (by induction) iff not XEZand not X E Y and X E f r ( r ) (by (iN iff not X E Y and X E f r ( r ) (since f r ( r ) s -Z) iff XE f r ( V y r ) . (vi) For every ?I, r : tut,r, r are compatible. Proof. Apply induction on deg. The statement is trivial for rEAt and carries through under the sentential operations. Consider now V y r and tufq(Vyr)=V,tut,rep,,F,y,r. Assume that V,vEpart(tut,V,r). If V,v= tutsVyr, then W = Z . Since fr(tut,V,r)=fr(V,r) by (v), XE fr(V,v) implies X E f r ( r ) . Now the choice of j ( r , q ) guarantees that Znpnrt(q(x))=O; hence W= 2 implies W n p a r t ( q (x)) = 0. If, on the other hand, V, v # tut,V', r, then V, vEpart(tut,rep,,r,y, r ) . But induction gives that tut,,repVcry , r, q are compatible. Hence x ~ f r ( V ~implies v) Wnpart(q(x))=O. At this stage, the desired substitution can be introduced. Namely, for every q, define the function sub, from u ( P ) into u ( P ) by sub,r=rep,r if r, q are compatible, and sub,r = rep,, tut,r otherwise. One obtains (vii) For every 'I, r, for every model A and every cp~u(A)':q ~ e ~ ( s u b , r ) ifh;qEe,(r). Proof. If r, q are compatible, this follows from (i). Otherwise e,(sub,r)= eA(rep,tut,r), eA(r)=eA(tut,r) by (iv). But cpEe,(rep,lut,r) iff h;qEeA(rutqr) by (i) since tut,,r and y are compatible by (vi).

EQUATIONAL MAPS

161

References 1. G. BIRKHOFF, On the structure of abstract algebras, Proc. Cambridge Phil, SOC.31 (1935) 433454. 2. P. M. COHN,Universal algebra (New York, Harper and Row, 1965). On the equivalence of certain classes of algebraic systems 3. B. CSAKANY(B. CAKAN), (Russian), Acta Sci. Math. Szeged 23 (1962) 46-57. On primitive classes of algebras which are equivalent to 4. B. CSLKANY(B. CAKAN), classes of half-modules and modules (Russian), Acta Sci. Math. Szeged 24 (1963) 157-1 64. 5. B. CSAKANY(B. CAKAN),On Abelian properties of primitive classes of universal algebras (Russian), Acta Sci. Math. Szeged 25 (1964) 202-208. On induction and recursion in universal algebra, to appear in: Z. math. 6. K. H. DIENER, Logik und Grundlagen d. Math. 7. W. FELSCHER, Adjungierte Funktoren und primitive Klassen, Sitzber. Heidelberg. Akad. Wiss., Math.-Natw. K1. no. 4 (1965) 1-65. 8. H. J. HOEHNKE, Zur Strukturgleichheit axiomatischer Klassen, Z . math. Logik und Grundlagen d. Math. 12 (1966) 69-83. uber Modellkorrespondenzen, to appear. 9. H. J. HOEHNKE, 10. C. R. KARP,Languages with expressions of infinite length (Amsterdam, North-Holland Publ. Co., 1964). 11. F. W. LAWVERE, Functorial semantics of algebraic theories, Dissertation (Columbia Univ., New York, 1963). Some aspects of equational categories, in: Proc. Conf. on Categorical 12. F. E. J. LINTON, algebra (Berlin-Heidelberg-New York, Springer-Verlag, 1966) pp. 84-94. 13. A. I. MALCEV, Structural characteristics of certain classes of algebras (Russian), Dokl. Akad. Nauk SSSR 120 (1958) 29-32. and A. TARSKI, The algebra of topology, Ann. of Math. 45 (1944) 14. J. C. C. MCKINSEY 141-191. 15. B. H. NEUMANN, Special topics in algebra, Universal algebra, Lecture notes (New York University, 1962). 16. J. SCHMIDT, Algebraic operations and algebraic independence in algebras with infinitary operations, Math. Japonicae 6 (1962) 77-1 12. Die Charakteristik einer allgemeinen Algebra I, Archiv d. Math. 13 (1962) 17. J. SCHMIDT, 457-470. 18. J. SCHMIDT, Some properties of algebraically independent sets in algebras with infinitary operations, Fundamenta Math. 55 (1964) 123-137. 19. J. SCHMIDT, Die uberinvarianten und verwandte Kongruenzrelationen einer allgemeinen Algebra, Math. Ann. 158 (1965) 131-157. The theory of abstract algebras with infinitary operations, Rozprawy 20. J. SLOMINSKI, Matematyczne 18 (Warszawa, P.W.N., 1959). 21. A. TARSKI, A remark on functionally free algebras, Ann. of Math. 47 (1946) 163-165.