Trends in LATTICE THEORY
J. C. ABBOTT, GENERAL EDITOR United States Naval Academy
Contributors GARRETT BIRKHOFF HENRY CRAPO and GIAN-CARLO ROTA SAMUEL S. HOLLAND, JR. GEORGE GRATZER
VAN NOSTRAND REINHOLD COMPANY NEW YORK CINCINNATI TORONTO LONDON MELBOURNE UNIVERSAL ALGEBRA by George Grlitzer
1. Introduction. During my first course in abstract algebra and even for a long while after that I had difficulty remembering what computational rules can be applied in the different branches of algebra: rings, integral domains, fields, division rings, and so on. I wished there were a branch of algebra where I did not have to remember any axiom systems. I did not know that there was one, and that it was called universal algebra, or abstract algebra (as if algebra were not abstract enough), or general algebraic systems. Universal algebra can be defined as the study of properties that such diverse algebraic systems as groups, rings, lattices, and algebras (over fields)have in common. Have we not just defined the empty set? Not quite. Each of these examples can be considered as a set with a family of finitary operations, and this is the defini tion of a universal algebra. A universal algebra ~{ is an ordered pair
173 174 TRENDS IN LATTICE THEORY ger n, 1 is an n-ary operation, that is, a mapping of An into A. In case n = 0, AD is the set whose only element is 0, the void set. Thus 1 is determined by 1(0). In other words, a nullary operation picks out an element of A. Examples of nullary operations are the o and 1 in any usual axiom system of Boolean algebras. For vari ous purposes it is very convenient to have nullary operations. We would not gain much by excluding them. Thus having convinced ourselves that the above definition of a universal algebra is the right one, there are two basic questions we have to answer. If we assume so little, as we do in the defini tion of universal algebras, is it still possible to develop a non trivial theory? And if the answer to the first question is in the af firmative, what has all this to do with a symposium on lattice theory and related subjects? It is the thesis of this lecture that the answer to the first question is indeed in the affirmative. I try to prove my case by re viewing what I consider the major accomplishments of this field, which is abundant in very deep results. There is a very close connection between universal algebra and lattice theory, partly for personal and partly for mathematical reasons. Professor G. Birkhoff, who invented lattice theory, was the first to publish non-trivial results in universal algebra. He is also responsible for popularizing the subject by lectures as well as by ample references to it in his widely read book on lattice theory. Thus it is not very surprising that quite a few mathematicians who started in lattice theory took interest in universal algebra (the pres ent author as one example) and vice-versa (B. J6nsson). There are lattice-theoretical results galore in universal algebra, and there are many results in lattice theory inspired by universal algebra (to which the major part of the last section is devoted). UNIVERSAL ALGEBRA 175
2. Basic concepts. If
is useful to have a fixed well-ordering of Fj F = sociate with 9f the sequence r =:: tions will be denoted by the same symbols fy' though (fy}9( and (fy}m would be more appropriate. If we talk about two or more alge bras, we will always assume them to be of the same type unless otherwise specified. Now the basic concepts can be defined. A homomorphism ¢> of the algebra ~l into the algebra 1a is a mapping of A into B which preserves all the operations, i.e., f/ao"'" any. I)¢> =:: f/ao ¢>"'" any_l ¢>}, for any ao'..., any _I f A and y < 0 (r). 1a is a subalgebra of ~ ~ if B A, and the operation fy of 1a is the restriction of the opera tion f of 9f to Bny, and B is closed under all the f ' A congru v y ence relation 6 of ~ is an equivalence relation on A satisfying the substitution property for all f ; that is, a. == b.(8}, i =:: 0, ..., y II nY - 1 implies fy(ao"'" any· I} fy(bo'"'' bny- 1}(8}. Polynomials (over ~O and polynomial symbols (of type r) also play an important role. Using the operation symbols f y' Y < Xl' ..., ' ... o (r), and the symbols xo' xn we build up symbols from the Xi by substituting them in operation symbols. Thus if no = 2, n l =:: 1, the following are examples of polynomial symbols: 176 TRENDS IN LATTICE THEORY XO' Xi' fO(xO' Xi)' f 1(fO(fO(XO' Xl)' f1(xO))) , and so on. If we use only xO"'" Xn -1 we get the tt-ary polynomi al symbols. Let ~ be an algebra, e an tt-ary polynomial symbol and a '"'' a _ € A. We can define p(a "'" a _ ) in a natural manner. o n 1 o n 1 Indeed, if p ". xi' let p(ao"'" an_ 1) == ai • If, e.g., P ". [l(fO(£O(Xo' Xl)' [1(x2))), then p(a a1' a ) ". [l(£O(£O(a a ), [1(a Thus p induces an n o' 2 o' 1 2»). ary function p over A, called a polynomial over~. The collection of all (of all tt-ary) polynomials over ~l is denoted by p(~n (p(n)(~». The set of all (of all n-ary) polynomial symbols is denoted by P(r) (p(n)(T». If p and q are polynomial symbols, then the expression p". q is an identity. The algebra ~{ satisfies p ". q if P and q induce the same function over A. These are the basic tools of an algebraist. And as we shall see in the subsequent sections, quite a bit can be accom plished using only these. These basic concepts can be defined with little or no changes for various generalizations of the concept of universal al gebra. Such generalizations consider partial operations, infinitary operations, relations, and so on. Of these, partial algebras are most useful in contributing to the theory of universal algebras. In this report, I will refrain from reviewing these other theories, but occasionally I will mention extensions of known results to the in finitary case. 3. From the Homomorphism Theorem to the Jordan-Holder Theorem. Once Kurosh remarked that there is no point in giving UNIVERSAL ALGEBRA 177 various generalizations of the concept of groups if the theory does not do more than extend the homomorphism theorem of groups. In deed, the homomorphism theorem holds for any algebra: every homo morphic image of ~{ is isomorphic to some quotient algebra ~/a (a homomorphic image 18 is any algebra such that there is a homomor phism from i"!l onto 18, and the quotient algebra ~(fa is the algebra defined in the usual way on the set of equivalence classes under e, where a is a congruence on ~{). Similarly, the two isomorphism theorems of groups hold for any algebra. The first states that there is a I-I correspondence between the congruence relations of ~ua and the congruence relations ell of ~ of which a is a refinement (i.e., x 5; y{a) implies x E y(eIl)), while the second states: where 18 is a subalgebra of ~r intersecting every congruence class modulo a, and eB is the restriction of a to B. The latter isomorphism can be strengthened to a rather use ful one. Let m be a subalgebra of ~(, a a congruence relation of ~r, and ell a congruence relation of m such that aB is a refine ment of ell. Let [B] a denote the union of the equivalence classes under a which intersect B. Then [B]a is closed under the op erations and it defines, therefore, a subalgebra [18]a of ~. On £B]a we can define the relation a(eIl) by the rule: x E y(a(eIl)) if there exist bo' bi l B with x E bo(e), bo bI (eIl), bi E y(a). Then a (ell) is a congruence relation on [.\8] a and The celebrated Zassenhaus lemma is a simple corollary of this isomorphism. All we have to do is to apply it twice. This proof of the Zass~haus lemma is as simple as, if not simpler than, 178 TRENDS IN LATTICETHEORY any known proof for groups or rings. The simplicity of the proof arises from the lack of any involved structure. This proof not only unified the known ones but helped to strengthen some of them, for instance, the Zassenhaus lemma for standard ideals of lattices. The Jordan-Holder theorem has no analogue for arbitrary algebras; some mild permutability condition on the congruences is inevitable. Several such extensions of the Jordan-Holder theorem 1 2 were proved by Goldie, Gould, the author and others. The author extended One of these to certain COncrete categories and recently Wyler to so called injective categories. The results mentioned in this section are rather simple minded. They have been included only to point out that universal algebra is the natural framework for several basic facts. 4. Algebraic constructions. One basic problem of algebra is to find methods of constructing new algebras from given ones. The best known and most thoroughly (though not satisfactorily) in vestigated construction is the direct product. Given the algebras ~ i' i £ I, we form the cartesian product A of the sets Ai' i £ I, and define the operations on A componentwise. The resulting al gebra ~l is the direct product oUhe algebras ~li' i £ I; in sym bols, m: = n(~ Ai £ n. It is not difficult to describe all direct pro duct representations of ~l in terms of congruence relations (Birk hoff). One of the most difficult problems is to determine which al gebras have the common refinement property, that is, if ~ has two direct product representations II(2( :I h n and II(m.! j £ J), then when 1 ) does ~ have a representation n(~ ..1i £ I, j £ J) such that ~. I) 3,4 4 1 '" II (~ ..Ij £ J) and ~. ~ II (~ ..Ii £ n? J6nsson, Tarski, Craw- l) s ) I) ley:Chang and others investigated this problem but even the sim- plest cases are full of unsolved problems. UNIVERSAL ALGEBRA 179 Many algebras (for instance non-atomic Boolean algebras) have direct decompositions but they do not have direct decomposi tions into directly indecomposable factors. A construction which does not have this defect is the following: A subalgebra ~ of II( ~{) i € I) is called a subdirect product of the ~li' i € 1, if for any given i € I and a. € A., there is an element b € B whose i-th II • component is aj Birkhoff's fundamental result states that every algebra is isomorphic to a subdirect product of subdirectly irreduci ble algebras. (A recent result of the author is that this theorem is equivalent to the Axiom of Choice. This is a solution of a problem proposed by H. and F. Rubin in their book, Equivalents of the Axiom of Choice, p. xv.) There are many other constructions which are associated with direct products, in that they give rise to subalgebras or homo morphic images of direct products. A subalgebra ~ of the direct. product II ( ~{j Ii € 1) is a weak direct product if for f € Band g € II( m.;1 i € I) we have that g € B if and only if Iilf(i) ,;, g(iH is finite. The algebras m. ;' i € 1 may have many or no weak direct products, but groups and rings always have exactly one weak direct product. Let L be a fixed ideal of the Boolean algebra of all su~ sets of 1. If in the previous definition we change the condition that iii f(i) ,;, g(i)J be finite to Ii I f(i) I: g(i)J € L, we get the concept of L-restricted direct product. A complete characterization theorem for L-restricted direct products in terms of congruence relations was 6 given by J. Hashimot0 . In terms of L, we can define a congruence relation eL on II( ~lil i l I) by the rule: f =" ) f(i) € g(eL if and only if iii l:,g(i)J L. 180 TRENDS IN LATTICE THEORY The quotient algebra IT( ilfj' i f [)L is denoted by ITL(ilfjl i f [) and it is called a reduced product of the ilf j' i f I. The most im portant special case is when L is a prime ideal. In this case ITL( ilf j' i f I) is called a prime product. This concept is due to 7 J. -t.os and it proved to be the most useful algebraic construction in the study of first order properties (see section 8). Direct limits and inverse limits can be defined for algebras the same way as for groups. Inverse limits are quite hard to visual ize. One exception occurs when the underlying partially ordered set is well-ordered. A recent result of the author reduces in a certain sense the construction of arbitrary inverse (direct) limits to well ordered inverse (direct) limits. The result states that if a class of algebras K is closed under isomorphisms and well-ordered inverse (direct) limits, then K is closed under arbitrary inverse (direct) limits. Let ilf be a homomorphic image under the homomorphism ¢ of the subdirect product ~ of the algebras ~f j' i f I. In a na tural way ilf induces a congruence relation ®j on ilf j . (x == Y(®j) if for a, bfA, the conditions that the i-th component of a is x and the i-th component of b is Y, imply a¢ = b¢.) If all the ®j are trivial (x == y(® j) implies x = Y), then I ilf j' i f II is a subdi rect covering of ilf. M. Yoeli found all subdirect coverings of an al gebra ilf in terms of a generalization of congruence relation (unpub lished result). This has interesting applications to automata theory. Now for a change, let ill be a finite algebra and ~ an arbi trary Boolean algebra. Let A[ ~] be the set of all functions a of A into B satisfying aa /\ ba = 0 for all a '" b and V (aa' a f A) = 1. If f is an n-ary operation of ilf we define f on A [~1 by HaO'""" an_I) = fJ, where fJ is given by UNIVERSAL ALGEBRA 181 af3 0: V(aoao A'" A an_ 1 an_II £(ao"'" an_I) 0: a) • The resulting algebra ~{[58 I is called the extension of ~( by the Boolean algebra 58 This construction is due to A. L. Foster.s Boolean extensions of finite algebras were characterized in S a special case by A. L. Foster and in the general case by M. 1. Gould and G. Gratzer. If ~ is infinite, 18 has to be lAI-complete for the above construction to work. However, no characterization theorem is known for this case. Each algebraic construction gives rise to an operator on classes of algebras. The most important ones are I, H, S, P, de fined as follows: for a class K of algebras I(K), H(K), S(K), P(K) are the classes of all isomorphic copies, homomorphic images, sub algebras and direct products of algebras in K, respectively. A set of operators generates a partially ordered semigroup. The finiteness of some of these semigroups was proved by E. Nelson and D. Pigozzl. A class of algebras K is called an equational class if K is closed under H, Sand P. Birkhoff's classical result (1935) states that a class K is an equational class if and only if K is the class of all algebras satisfying a set of identities. (The proof is via free algebras; see section 6.) 5. Related structures. Suppose from a given algebra ~( (from a class of algebras K) we constructsome new structure, say 18. This ID may be e.g., 'a set of positive integers, or a lattice, or a group, or a topological space. 58 may be anything which reveals something interesting about il! (or K). The general problem of characterizing what sort of m we get is central in mathematics and certainly provides the most interesting problems in universal algebra. While in algebra most of these problems are almost impossible to at tack (e.g., characterize the lattices of normal subgroups of groups) 182 TRENDS IN LATTICE THEORY in universal algebra they provide an endless list of interesting and very seldom hopeless, though sometimes very hard problems. (a) Systems of sets. For an algebra ~(, let S denote thesys tem of subsets B of A such that (if) if a f [xl, then a f [Xl], for some finite Xl ex. Systems of subsets satisfying (i) and (if) are called algebraic clo- sure systems. Various generalizations of this result are known. FOr exam ple, it has been generalized by the present author to infinitary alge bras and to systems of subalgebras (of an (infinitary) algebra) gen erated by fewer than a certain number of elements, and it has also 10 been generalized, by O. Frink and the author, to closed subalgebras of topological algebras. A known unsolved problem of this kind is the characteriza tion of the system of independent subsets of ~( (see section 7). The multiplicity type fl. of the algebra ~(= THEOREM The following four conditions on the lattice 2 are equivalent: (i) 2 is an algebraic lattice; (ii) 2 is isomorphic to the subalgebra lattice of some algebra ~( ; (iii) 2 is isomorphic to the subalgebra lattice of ~r x ~r for some algebra ~r; (iv) £ is isomorphic to the congruence lattice of some algebra ~. The equivalence of (i) and (ii) is due to Birkhoff and Frink; 9 it follows trivially from the result in (a) and very simple direct proofs are known. The equivalence of (i) and (iv) is due to G. Grat zer and E. T. Schmidt; the only known proof is rather complicated. It would be of some interest to obtain a direct proof. Condition (iv) is due to A. A. Iskander.12 (c) Groups and semigroups. The endomorphisms of an alge bra form a semigroup, called the endomorphism semigroup, and the 184 TRENDS IN LATTICE THEORY automorphisms form a group called the automorphism group. A semi group & is isomorphic to the endomorphism semigroup of an algebra if and only if it has an identity element (the algebra can be taken as the set E with the left multiplications as unary operations), and any group is isomorphic to the automorphism group of some algebra. Let ~ be the endomorphism semigroup of 9f. Let ~ 0 and ~1 be the subsemigroups of onto and 1-1 endomorphisms, respec 13 tively. M. Makkai solved the problem of characterizing the triplet <~'~O'~l>' This is a highly non-trivial result. (d) Combinations of (a)-(c). E. T. Schmidt proved that the automorphism group is independent of the subalgebra lattice. To state this result precisely, let The author proved that the congruence lattice and the endo morphism semigroup are dependent. The exact nature of this depen dence is not known. (e) Sequences. Let K be an equational class and let Sp(K) (the spectrum of K) be the set of finite cardinalities of alge bras in K. The following two properties of S :: Sp{K) are obvious: (i) 1 £ S; (ii) a, b £ S imply a' b £ S. The converse of this was proved by the author: given any set S of natural numbers satisfying (i) and (ii), there exists an equational class K with S:: Sp(K). The spectrum S of an equational class K defined bya finite set of identities has to be recursive (Asser, Mostowski). No char- " UNIVERSAL ALGEBRA 185 acterization is known for this case. However, a simple construction due to the author and a result of Higman and B. H. Neumann yield that, for any equational class K defined by a finite set of identities, there exists an equational class K1 defined by two identities such that Sp(K) == Sp(K t ). We can consider Sp(K) as an (U-sequence THEOREM. The closure of B in 2No is A. ~~ Let ~{ be quasi-finite if IP(n)(ml < NO for all n < (U. IP(n)(~l)1 ~n ~nl, p(n)(~o Set cn == and == IP(n)( where is the set ~{ of n-ary polynomials over depending on every variable. Let CK ~O' ~l"'. and CK denote the set of all sequences (iv) if (Pi = q) ( ~, then (f/po"'" Pn -1) = fy(qo"'" qn -1)) y Y (2: for f ( F; and y (v) if p = q is in ~ and we get p' resp.q' by replacing all occurrences of xi by a polynomial symbol r in p resp. q, respectively, then (p' = q') f ~. The proof uses the fact that &K(m) exists for all m if K is equa tional, and that, for the equational classes K and K 1 the following conditions are equivalent: ; (a) K S;; K1 (m Id(K)::2 Id(K1) ; (y) &K«i)) is a homomorphic image of &K «i)). 1 Thus every problem of free algebras can be stated as a prob lem on identities, and vice-versa. We will always use the formulation which is simpler. Thus we see that free algebras are algebraic equivalents of closed sets of identities. This leads to some interesting definitions. For instance, we can take a minimal equational class K (i.e., if Ko c K, Ko equational, then Ko contains only one element algebras) and we call Id(K) equationally complete. Every set of identitiesthat UNIVERSAL ALGEBRA 187 can be satisfied by an algebra with more than one element can be extended to an equationally complete one (A. Tarski) and there are It 2 0 equationally complete sets of identities for any finite type con- taining at least one binary operation O. Kalicki).!4 Set K "" HSPO ~1 D. Then Id( ~1) "" Id(K). One would sup pose that the structure of Id( 90 is very simple if 9( is finite. Un fortunately, this is not so. R. C. Lyndon proved that Id( 90 may not have a finite basis. (A set of identities, ~ has a finite basis if for some finite ~ I £ ~ we have that ~ is the smallest closed set of identities containing ~ 1)' 8 A. L. Foster's primal algebras (9£ is primBI if A is finite and for each n, every n-ary function is a polynomial) are examples of algebras for which Id(~) always has a finite basis (A. Yaqub). An important property of the primal algebra ~£ is that every ~ f K is isomorphic to a subdirect power of 9( and if ~ is finite, then ~ is isomorphic to a direct power of 9£ (L. I. Wade). Vari ous proofs and generalizations of this result have recently been dis covered.(A. L. Foster, Pixley, Astromoff). The celebrated word problem also found its way into univer sal algebra. For a finite set of identities ~ the word problem is solvable if for every finite system tPi "" qilUI and p "" q, thereis an effective process that decides,if whenever m; is an algebra sat- ~, isfying BO'"'' Bn _ 1 f A and", (ao"'" an_I) "" qi (ao,m, an_I) for all i f 1, then p(ao,..., an_I) "" q(ao'"'' an_I)' T. Evans15proved that the word problemJor k is solvable iff for every finite partial algebra ~£ it is decidable whether ~( canbe embedded in an algebra satisfying k. Free algebras are closely connected with another important algebraic construction, namely the free product (R. Sikorski!6). The 188 TRENDS IN LATTICE THEORY algebra \}{ is a free product of the algebras \}{i' i f I, over a class K of algebras if there exist embeddings (1-1 homomorphisms) ¢ i of \}{. into \}{ such that A is generated by the set U(A . f I) and I I 'f'I-J...I i whenever ~ f K and .1•. , f I, are homomorphisms of \}l. into ~, 'f', i 1 then there exists a homomorphism ¢ of I!( into ~ with ¢ i ¢ '" if; i for all i f I. 'tY K(Ul) is always a free product of m copies of 'tY K(I). Free algebras 'tY K(rn) can be easily characterized by the property that the identities which hold on the basis elements must hold in K. A similar, but more involved, «logical" definition of free products was given by J. 1.os.17 It seems very unfortunate that so little is known about free products. A satisfactory theory of free products could advance quite a few chapters of universal algebra. 7. Bases of free algebras. A vector space is always free, and a basis in the usual sense is the same as a basis in the sense of § 6. The first result one proves for a vector space is that any two bases have the same power. This is not true in free universal algebras. This problem arises, however, only with finite bases, since if one basis is infi nite, then all are infinite and have the same power. Let DCI!() de note the set of cardinalities of all bases of the free algebra \}l. E. Marczewski proved that if DC I!() contains more than one element, then DC\}O is an arithmetic progression. And, conversely, every arithmetic progression can be represented as D( \}() for some \}{. This was proved in a special case by Goetz and Ryll-Nardzewski and in its full generality by Swierczkowski~8Leavitt improved this result by showing thatl}{ can always be chosen as a module. UNIVERSAL ALGEBRA 189 These results were extended recently to infinitary algebras by Burmeister and the author. E. Marczewski was the first to emphasize that not only the independence of vectors, but almost all the independence concepts in mathematics are special cases of the "independenceH of ele ments of a basis. More formally, let ~l be an algebra, K =: HSP ( ~n, the equational class generated by m:. Then the set 1 of elements of'll is independent if the subalgebra ~ generated by 1 is free over K and 1 is a basis of ~. This does not, at first glance, strike one as a very fruitful definition. Nevertheless, it resulted in the discovery of quite a few interesting (and a few brilliant) results. Many results can be readily formulated using the follOWing six constants which were first systematically discussed by Marczew ski. In the first half of this section let 'll be a finite algebra. g*('lO =: the smallest integer n, such that every n-element subset of A generates'll. g(m:) =: the cardinality of the smallest generating system of m:.. i ('ll) =: the cardinality of the largest independent subset of m:.• i*(~) =: the maximal n such that all n-element subsets of A are independent in ~l. Call an n-ary polynomial trivial if it equals one of the pro ) =: Xi' jection functions e:, defined by e:(xo,...,xn_ 1 Then 190 TRENDS IN LATTICE THEORY 00, if all polynomials over 2l are trivial, n , if n is the largest number such that all n-ary polynomials over 2l are trivial, 0, if there are no constant polynomials, p(2l) == and at least one non-trivial unary poly nomial, -1, if eitherlAI == 1 or 2l has at least one constant polynomial. These numbers are invariant not only under isomorphism but 1> also under equivalence (21 == (iii) If g* == g == i, or g == 1 == i*, then g* == g i == i*. (Marczewski) (iv) If i* == IAI > 2, then 21 is trivial. (Swierczkowski) (v) 21 is free means that i == g. A graduate student at Pennsylvania State University, G. W. Wenzel, gave a complete solution of the characterization problem of the 15 possible pairs of the above constants. His results are sum marized in the following table: UNIVERSAL ALGEBRA 191 U ! Of these, the cases 13) and 14) are the hardest. Wenzel also got some interesting results on the pairs of con stants of free algebras as well as a complete description of the six tuple for free unary algebras (unary means that in the type ...> all n j are 0 or 1). Typical results are the following: 1 1 (i) Let ~l be a free algebra with p =: 2, i* =: 3. Then 2 - divideslAI· 192 TRENDS IN LATTICE THEORY (ii) Let ~{ be a free algebra with p ~ 3. Then i divides IAI. '<\,J;- The notion of independence is also used to define new, in teresting classes of algebras by the properties of independent sets. A class of algebras in which independent sets behave very much as in vector spaces is the class of v-algebras (E. Marczewski): ~{ is a v-algebra if whenever p, q ( p(n)( ~O, and p = q can be dis tinguished by x _ , then for some r ( p(n-l)( ~O, p(aO'·oo, an_I) n 1 = q(ao,.oo, an-1) iff a.1-1 = r(ao'··o, an- 2)' for all ao"'" an-1 ( A. (p = q can be distinguished by x _ means that there exist n 1 bo'·'·' bn-l' b'n-1 ( A with p(bo'··o, bn-1) = q(bo"'" bn-1) and p(bo'···' b:_1) ~ q(bo'''·' b:_1))· In v-algebras, independent sets have the usual exchange property, every v-algebra has a basis, and all bases have the same cardinality (the dimension). Urbanik19 shows that every v-algebra which has a basis of more than two elements of dimension ~ 3 is equivalent to a vector space! A v*-algebra ~ is defined by the following two properties: (i) if a ( A is not the value of a constant polynomial, then {al is independent; (ii) if n > 1, {aO'"'' an_II is independent, and {ao"'" an_1' a is dependent, then a ([a "." a 1] (the subalgebra n I n o n- generated by {ao"." an_ 1 D. A v**-algebra ~r is one in which the condition: ai I [ao'"'' ai_I' ai + 1"'" an_I], i = 0, ..., n-1, implies that {ao'"'' an_II is independent. All v*-algebras have bases but this is not true of v**-alge bras. Nor does the exchange property hold for independent sets. UNIVERSAL ALGEBRA 193 Therefore, it is very surprising that if a v** -algebra has a basis, then all bases have the same cardinality (W. Narkiewicz). This is the first instance of a result of this kind in a situation where we do not have any exchange property! The representation problem of v-algebras is completely solved (K. Urbanik). A representation theorem was also given for v* algebras by K. Urbanik. This is not completely satisfactory since it is partially based on the notion of quasi-fields (introduced by the author, the name is due to K. Urbanik) about which very little is known. From this point of view v**-algebras are very bad; so many pathological examples are known (due to W. Narkiewicz, K. Urbanik) that a characterization theorem seems rather hopeless. Finally, a rather difficult open problem should be mentioned, namely the characterization of the system of independent subsets of an algebra. Partial results are known (S. SWierczkowski, K. Ur banik, S. Fajtlowicz 20) but the complete solution seems to be as yet unattainable. 8. The first order language of algebras. For a given type r, starting with the identities p =" qt using the logical connectives nandt" "or," "not;t (in symbols'" tV,-':) and the quantifiers "for all x. t" "there exists an x. t> (in symbols (x.), 3x.), one can build 1 1 1 1 up the formulas of a language L(r) t called the first order language associated with the type r. This is a first order language since one can quantify only the variables xi' A sentence is a formula in which each occurrence of every vari- able is bounded by a quantifier. It is intuitively clear what it means to say that then cI>(a) holds for aU elements of a group A formal definition of "satisfaction" was given by A. Tarski 21. A sentence If a sentence I i I• can be ordered, then Lowenheim-Skolem-Tarski Theorem 21• Let ~( be an algebra and H.£: A such that H is infinite and has at least as many ele ments as there are operations in ~. Then there exists an elemen tary subalgebra ~(1 of ~{ with H s;;: A 1 and IHI '" IAIl· These concepts can be used to give an algebraic characteriza tion of axiomatic classes: K is an axiomatic class if and only if K is closed under isomorphism, and under the formation of prime products (see §4) and elementary subalgebras. Similar algebraic characterizations can be given of various other classes. These are not hard to prove. The following result, however, is very deep (H. J. Keisler 2~: Let us assume the Generalized Continuum Hypothesis. Then \1{ and ~(1 are elementarily equivalent if and only if they have isomor phic prime powers (i.e., prime products each of whose factors is equal to ~( and ~(1' resp.). Given a formula lIl, it can always be transformed into a formula which has all the quantifiers at the beginning (this string of quan tifiers is called the prefix) and these are followed by a formula without quantifiers (called the matrix). Such formulas are said to be in prenex normal form. The formal properties of a prenex normal formula are very im portant. For instance, if III is a sentence in prenex normal form, and all quantifiers are universal (a so-called universal sentence), then whenever III holds in ~{, III also holds in all subalgebras of ~L In other words, all models of III are closed under the formation of subalgebras. For instance, if a ring is commutative, then so is every subring. For almost every algebraic construction, it is easy to find for mal properties of III which guarantee that if III holds in K, then it UNIVERSAL ALGEBRA 197 will hold in any algebra which we get from K by performing that algebraic construction. Let us list a few examples, starting with the above one. (a) Universal sentences are preserved under the formation of subalgebras. (b) Positive sentences are preserved under the formation of homomorphic images. ($ is positive if the matrix does not contain the negation sign.) This is very natural, since -r (xci = x 1)' i.e., xo ;' Xl' should not be preserved under homomorphic images, but everything not in volving.-, should be preserved. (c) 'Q'3 -sentences are preserved under the formation of directed unions of algebras. (A 'Q'3 -sentence is one with a prefix in which no universal quantifier follows an existential quantifier, e.g., (x )( 3xlh/t, (3x (x where is the matrix. A di o 2)l/t, 3)l/t, l/t rected union is a direct limit in which all homomorphisms are one-to-one.) (d) Horn sentences are preserved under the formation of reduced direct products. (A Hom sentence is defined by the pl'operty that the matrix of is a conjunction of formulas of the form l' ®oV ... V ®n_ where the ®,1 are identities or negations of identities, but at most one ®2 1 is an identity. Example: (Xo)(X 1)(x2}(xOXI ;, xOx 2 V xl = x2), or in more usual form: xOx l = xOx 2 implies x I = x2' which is a can cellation law.) (e) Conjunctions of identities, considered as universal 198 TRENDS IN LATTICE THEORY sentences, are preserved under the formation of subalgebras, homomorphic images and direct products. Of these, (a), (b) and (e) are trivial, (c) is easy to prove and (d) is the only non-trivial statement (C. C. Chang and A.C.Morel23), We cannot really expect that the converse statements of (a)-(e) hold. For instance, <1>: (x )(x )(:3 x )(x x =x x V x .;, x ) is O 1 2 O l 1 O 2 2 not universal, but it is preserved under the formation of subalgebras. The reason is obvious: is equivalent to (x )(x )(x x >= x1x ), O 1 O l O which is universal. (The sentences and 1 are equivalent if holds in the algebra! iff <1>1 holds in !.) This gives us the clue: we want the converse of (a)- (e) only up to equivalence of formu las. And the converse up to equivalence of formulas are indeed true statements. They were proved by the following mathematicians: (a) J. -Los and A. Tarski; (b) J..{:,os, A.I.Mal'cev, R.C.Lyndon; (c) C. C. Chang, J. -Los, R. Suzko; (d) H. J. Keisler 24; (e) G. Birk hoff. Of course, (e) is nothing but a new form of a result already men tioned twice (§§4 and 6). The deepest by far is (d), the proof of which combines the technique of the algebraic characterization of elementary equivalence with an intimate knowledge of the "special models" ofM. Morley and R. L. Vaught. ~,);> A further topic should be mentioned, even if nothing specific will be said about it. This is the decidability problem of first order theories, which has been very extensively discussed. However this is based on recursive function theory, which we do not ~ v Taimanov, M. A. Taiclin) is available. UNIVERSAL ALGEBRA 199 9. Free "i.-algebras 25. In this section "i. will always stand for a set of first order sentences written in prenex normal form; a mod elof "i. will becalled a "i.-algebra. We know from §8 that in general the formation of subalgebras and homomorphic images takes us out of the class of "i.-algebras. The problem arises as to how we could introduce stronger concepts which do not have this defect. Sometimes a very simple trick helps. Let "i. be the usual axiom system of groups (*) Then the subalgebras are only subsemigroups with identity. But by the introduction of -1 as a unary operation, changing the type from <2, 0> to <2, 1,0>, the troublesome axiom (*) can be transformed into a universal sentence: Now subalgebra is the same as subgroup. Two questions arise: (i) Could we not sQmehow define a sub algebra concept which in the above example gives us the sul:groups, without changing the type? (ii) Does the trick illustrated above ("introduction of Skolem-functions") always work, and if not what c an we do then? It is easy to provide an example which answers (ii): take for "i. an axiom system of complemented lattices It is obvious that there is no "natural" way in which the comple- 200 TRENDS IN LATTICE THEORY ment can be introduced as a unary operation (which complement?). The way we will solve the problem by (ii) will also answer (i). If there is no way to select a single complement (in fact all complements may be absolutely symmetric), then if we want at least one in the subalgebra, we must put in all. This leads us to the definition of I-subalgebras, the formal definition of which will not be given here. Intuitively, if 2£ is a I-algebra and 2f is a t subalgebra of 9f, then ~£t is a I-subalgebra if all Ilinverses" in 2f of elements of A are also in At (the Hinverses" are those l elements of A guaranteed to exist by I). (To define a I-subalgebra as a subalgebra satisfying I would be very unsatisfactory: the intersection of I-subalgebras would not be a I-subalgebra in general.) Examples: If (1) a o ( At' and t/r(ao> at) in A, then at ( At' If (2) a (At' at ( A, and (x )t/r(a ' at' x ) in ~, then at ( At" o 2 O 2 The general definition of I-subalgebra is too technical to be stated here, but it is easy to imagine that all we need is a good definition of a I-inverse (to be given by induction on the number of existen tial quantifiers in the prefix) and then a I-subalgebra ~{t of ~( is a subalgebra closed under the formation of I-inverses in ~ of ele ments of At' UNIVERSAL ALGEBRA 201 A similar argument shows that a I.-homomorphism should bede fined as a homomorphism ¢: A .... B which preserves the inverses, and such that the images of the elements of A have no more in verses than those we get by ¢. If ~{ is a I.-algebra and H s;;; A, there exists a smallest I. subalgebra ~ containing H. We will say that H I.-generates ~. Now we have all the ingredients to define a free I.-algebra: the free I.-algebra lY I.(m) on m generators is a I.-algebra ~r, I. generated by a set H of cardinality m, with the property that any mapping of H into any I.-algebra }8 can be extended to a I.-homo morphism. Alas, this extension is not unique and this makes the theory of free I.-algebras much more involved than the theory of free alge bras. The uniqueness of free algebras is a simple consequence of the fact that any mapping of the basis has a unique extension to a homomorphism. So it is quite surprising that the uniqueness of free I.-algebras is still true. Uniqueness Theorem. For given I. and m, lYI.(m) is unique up to isomorphism. Some further typical results are the following: If lYI.(m) exists, then lYI.( n) exists for all n < 111 • If tYI.(~ exists for all n < cu, then lYI.(cu) exists. If lYI.(cu) exists, then lYI.(m) exists for all 111. Necessary and sufficient conditions for the existence of 0: I.(m) are known, as well as a rather complete answer to the question of when one can reduce the problem of existence of lY I.( m) to the 202 TRENDS IN LATTICE THEORY existence of some 3K( m}. These are too technical to be stated here. However, the whole theory is full of very basic unsolved prob lems. Let me illustrate one possible direction in which work seems desirable. Given an axiomatic class K, one can find many sets of sen tences :£. such that K is the class of all :£. -algebras. Let us say I I that :£0 is better than :£ 1 if :£0 allows more subalgebras and homo- morphisms. Is there always a best :£.? This gives a possibility of I comparing various axiomatizations of an axiomatic class, and dis- cussing the structure of all axiomatizations. 10. Some general comments and lattice-theoretic particulars. The simplest way to teach a child what animals are is to take him to the zoo. However, it must be remembered that relatively few ani mals are represented there. The same comment applies to this sur vey: the results given in §§ 3 -9 are in a sense arbitrary, and defi nitely they do not give a full picture. Nor does my book. There are at least 400 papers on universal algebra and even if we take away 20% (the trivial papers), the remaining .320 contains so much infor mation that nobody can really claim any more that he knows the sub ject thoroughly. Nevertheless, it is hoped that this survey, in its sketchy way, will help one to understand what universal algebra is. What is the connection of universal algebra with specific alge braic theories? Besides providing a simple framework for elemen tary results(§ 3) it made two major contributions to almost every field of algebra: (i) raising problems in the specific fields which would not have been asked otherwise; (ii) prOViding some general, and highly non-trivial, techniques which yield interesting results. UNIVERSAL ALGEBRA 203 As for (i), it is enough to remark that the theory of equational classes of groups became so extensive that it was necessary to write a book to survey the results. This was done by H. Neumann26• Similar but less successful attempts have been made in ring theory. As for (ii), the best example is the theory of prime products. This theory already has non-trivial applications in rings (A. Amitsur). diophantine equations (J. Ax and S. Kochen), field theory,and so on. lt took almost ten years for such an easy construction as prime products to infiltrate some chapters of algebra. A possible explana tion is that only a few algebraists took the trouble of investigating what results from universal algebra can be applied in their own fields. For instance, I think that the "special models" of M. Morley and R. L. Vaught will prove to be at least as important in applica tions as prime products. Let us hope that some day special models will be at least as well known as prime products are today, and then, I have no doubt, it will provide a useful tool for algebraists. <;(,!A' The connections between lattice theory and universal algebra are twofold. Lattice theory helped universal algebra with many characterization problems (§S) but there are less obvious connec tions. For instance, a very neat way of describing the equational class K generated by a given class K owas found by B. j:0nsson but this works only when all algebras in K have distributive con gruence lattices. Discussions of direct decompositions and common refinement theories usually end up in characterizations involving special classes of lattices. 204 TRENDS IN LATTICE THEORY It can be safely said that lattice theory is the best friend of universal algebra. The converse, however, is not true. But I think universal alge bra has made enough contributions to lattice theory to qualify as a good friend. Algebraic lattices were discovered in universal algebra and they proved to be a most interesting class from a purely lattice theoretic point of view, worth a thorough investigation (P. Crawley, R. P. Dilworth and others). Among the most recent developments in lattice theory I will pick out two problems which came from universal algebra, and in which I am most interested, to suggest further work in this area. (i) Equational classes of lattices. Let K be an equational class of algebras; then the equational classes Ko ~ K form a lat tice 2 K under inclusion. (Those who want to base their studies on axiomatic set theory will frown upon this statement, since a proper class cannot be an element of a set. Fortunately, £ K can also be defined as the dual of a sublattice of the congruence lattice of ~K(w), so everything is all right.) It is easily seen that £ K is the dual of an algebraic lattice the congruence lattice of ~ K(w) is distributive and £ K is also distributive (B. J 6nsson). We will be interested here in the special case where K is the class of lattices, and we will write 2 for 2 K• The zero 0 of 2 is the class of one element lattices. 2 has exactly one atom, namely D, the class of distributive lattices, and every A f L other than 0 contains D. (This is a restatement of G. Birkhoff's classicalresult: every distributive lattice is isomor phic to a subdirect product of two element lattices.) Let We 5 and in5 denote the lattices of Fig. 1, M5 and N5 the equationalclasses UNIVERSAL ALGEBRA 205 generated by them, It is known that every non-distributive lattice contains a sublattice isomorphic to9Rs or Ws' Thus Ms and Ns Figure 1. cover 0, no other element of L covers 0, and if A (L and A> D, then A contains Ms or Ns ' These elementary observations are shown in Figure 2. Figure 2. B. J6nsson raised the following problem: how many classes of modular lattices cover Ms ? It is easy to see that if A covers Ms' then there is a subdirect ly irreducible lattice generating A. The present author proved 27 206 TRENDS IN LATTICE THEORY that the only subdiredly irreducible finite modular lattices which generate a class covering Ms are those shown in Fig. 3. Figure 3. It is not known whether there is an infinite lattice with these properties. The proof of the theorem stated above is essentially a painstaking computation of free lattices over weird classes of lat tices. A less computational proof would be highly desirable. One can also ask the question of whether the identities of IDls' ~ Wl , andWl have finite bases. Since these are lattices, it is s' 6 g known that finite basis means one identity. Claims have been made by various mathematicians that the single identities for IDes and ~ s have been found, but proofs have never been published, nor com municated orally to me. (ii) First order properties of the lattice of ideals. The best known algebraic construction in lattice theory is the construction of S( 2), the lattice of all ideals of the lattice 2. The topic of §8 suggests the following problems: UNIVERSAL ALGEBRA 207 Find all first order properties cI> of lattices such that (1) ~ has cI> implies that ~ (~) has cI>; (2) ~ (~) has cI> implies that 2 has cI>; (3) 2 has cI> iff ~ (2) has cI>. The best known example for (3) is a conjunction of identities. One can, however, compile long lists of properties satisfying (1), (2), or (3), ranging from the trivial ones (ILl = n < w) to more tricky ones. The longer the list became the less hopeful I felt about getting a decent solution for (1)- (3). So I chose a particular class of prop erties which seemed to have special importance in lattice theory, For a finite lattice 91, let cI>( 9n denote the property that the lattice has no sublattice isomorphic to In. cI>( In) is obviously first order and it always satisfies (2). Thus (1) and (3) are equivalent and can be restated as follows: Find all finite lattices In with the property (P) that for any lattice 2, if ~ (~) has a sublattice isomorphic to In, then ~ has a sublattice isomorphic to In. A trivial example is 9C ' since cI>())(s) holds for 2 if 2 is S modular, and 2 is modular iff J' (2) is modular. Thus illS has (P). Here are some less trivial examples: ID?s does not have (P) (that is, one can construct a lattice 2 which has no sublattice ~ms)' isomorphic to 9JlS but 3 (2) has a sublattice isomorphic to G. Bruns pointed out that the statement on 9Cs can be general ized as follows: Let N be any finite lattice which is constructed from two finite chains, each of more than two elements, by identify ing both zeros and both ones. Then N has (P). 208 TRENDS IN LATTICE 1HEORY A. G. Waterman observed that the eight element Boolean algebra has property (P). In conclusion let me state the only general theorem I have found: If there is an element in 9t which is join and meet reducible, then 9t does not have property (P). This report (excepting the last section) is based on the author's forthcoming book on universal algebra, the first draft of which was mimeographed in 1964-1965 as lecture notes of Math. 572 at the Pennsylvania State University, and the second draft of which has just been completed. It is hoped that it will appear early in 1967 and the reader is referred to it for a more detailed development and a complete bibliography of the subject. Some of the research reported above was part of a research pro ject sponsored by the National Science Foundation under grant num ber GP-4221. I wish to express my gratitude to my students, M. I. Gould, C. R. Platt, R.M. Vancko and G.H. Wenzel who read the manuscript of this report, and contributed several useful suggestions to it. The Pennsylvania State University and The University of Manitoba UNIVERSAL ALGEBRA 209 FOOTNOTES 1. Proc. London Math. Soc. (2) 52 (1950), 107~131. 2. Magyar Tud. Akad. Mat.Kutata Int. Kozl. 8(1963), 397406. 3. Pacific J. Math. 14 (1964), 797-855. 4. Notre Dame Math. Lectures 11:5. 5. with B.J6nsson and A. Tarski, Fund. Math. 55 (1964),249~281. 6. Osaka Math. J. 9 (1957), 87~112. 7. Mathematical Interpretation of Formal Systems, 98-113. 8. Math. Z. 58 (1953), 306-336. 9. Trans. Amer. Math. Soc. 64 (1948), 299-316. 10. Archiv. Math. 17 (1966), 154~158. 11. Acta. Sci. Math. (Szeged) 24 (1963), 34-59. 12. Izv. Akad. Nauk SSSR, Ser. Mat. 29 (1965), 1357~ 1372. 13. Acta Math. Acad. Sci. Hungar. 15 (1964), 197-307. 14. Nederl. Akad. Wetensch, Proc. Set A 58 (1955), 660-662. 15. J. London Math. Soc. 28 (1953), 76~80. 16. Fund. Math. 39 (1952/53), 211-228. 17. "The Theory of Models," Proceedings of the 1963 Interna- tional Symposium, Berkeley, 229-237. 18. Fund. Math. 50 (1961), 3544. 19. Fund. Math. 48 (1959/60), 147-167. 20. ColI. Math. 14 (1966), 225~231. 21. Compositio Math. 13 (1958), 81-102. 22. Nederl. Akad. Wetensch. Proc. Set A 64 (1961), 477495. 23. J. Symb. Logic 23 (1958), 149-154. 24. Trans. Amer. Math. Soc. 117 (1965), 307-328. 25. Results announced in Magyar Tud. Akad. Mat. Kutatalnt. Kozl.8 (1963), 193-199. details are published in Trans. Amer. Math. Soc., 1969. 210 TRENDS IN LATTICE THEORY 26. Varieties of Groups, Ergebnisse der Mathematik, Springer Verlag, Berlin-West, 1967. 27. Duke Math. ]., 1966.