Trends in LATTICE THEORY -...

Trends inLATTICE THEORY

J. C. ABBOTT, GENERAL EDITORUnited States Naval Academy

Contributors

GARRETT BIRKHOFF

SAMUEL S. HOLLAND, JR.

HENRY CRAPO andGIAN-CARLO ROTA

GEORGE GRATZER

VAN NOSTRAND REINHOLD COMPANYNEW 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 <A; F> where A is a nonvoid set and F is a family of fini

tary operations on A. In most cases F is finite. However if, for

example, we want to consider a vector space as a universal algebra,

then F will have to consist of the binary addition and of the set

of unary operations fa(x) = ax, one for each element a of the un

derlying field. Thus if the field is infinite, so is F. If ~{ = <A; F>

is a universal algebra and f f F, then for some non-negative inte-

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 <Rj+, •> is a ring, then + and, are

both binary operations, however different in nature. To make sure

that we do not confuse them we use different symbols to indicate

these operations. Similarly, in a universal algebra 9f = <A,' F> it

is useful to have a fixed well-ordering of Fj F = <fo"'" fy'"'' >y<f3'Then the indices will serve as "names" for the operations. We as-

sociate with 9f the sequence r =:: <no"'" ny'''' >y<[3' called the

type of 2(, where ny is the "arity" of fy ' [3 will be denoted by

o (r). Given two algebras 9f, 1a with the same type r, the opera

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 fv

of 9f to Bny, and B is closed under all the fy' A congru

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 I I

n - 1 implies fy(ao"'" a I} fy(bo'"'' b 1}(8}.Y ny· ny-Polynomials (over ~O and polynomial symbols (of type r)

also play an important role. Using the operation symbols f y' Y <o (r), and the symbols xo' Xl' ... , 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:


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

ao'"'' an_ 1 € A. We can define p(ao"'" an_ 1) in a natural manner.

Indeed, if p ". xi' let p(ao"'" an_ 1) == a i • If, e.g.,

P ". [l(fO(£O(Xo' Xl)' [1(x2))),

then p(ao' a1' a2) ". [l(£O(£O(ao' a1), [1(a2»). Thus p induces an n

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


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,


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 theorem1 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 when1 )

does ~ have a representation n(~ ..1i £ I, j £ J) such that ~.I) 3,4 4 1

'" II (~ ..Ij £ J) and ~. ~ II (~ ..I i £ n? J6nsson, Tarski, Craw-l) ) I)

ley: Changsand others investigated this problem but even the sim-

plest cases are full of unsolved problems.


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-thI I

component is a j • 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 ( ~{jIi € 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

given by J. Hashimot0 6.

In terms of L, we can define a congruence relation eL on

II ( ~lil i l I) by the rule:

f =" g(eL ) if and only if iii f(i) l:,g(i)J € L.


The quotient algebra IT( ilf j ' 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

J. -t.os7and 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


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 inS

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)

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 <B;F> is a subalgebra. Westip

ulate that 0 is in S if and only if there are no nullary operations in

F. S is called the subalgebra system of ~(. S is characterized9

(Birkhoff and Frink) by the following two properties:

(0 S is a closure system, that is, S is closed under arbitrary

intersection (by (i), for X S; A, there is a smallest member

[xl of S containing X);

(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 also10

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 ~(= <A; F> is a se

quence < mO' ml ,... ,mn, ... >, where mn is the cardinality of the set

of n-ary operations of ~(. Let T (fl.) be the class of subalgebra sys

tems of algebras of multiplicity type fl.. Then fl. == v iff T(fl.) ""

T(v) defines an equivalence relation on the multiplicity types. Let

; denote the equivalence class containing fl.. Write ;:.s;; if T (fl.)

S; T(v). M. 1. Gould (a graduate student at Pennsylvania State Uni

versity) described the relation; :.s ;; and also found a "normal-

form H theorem; i.e., he found a class of multiplicity types which

intersects each equivalence class in exactly one element.

(b) Lattices. An element a of a complete lattice 2 is

called compact if a :5 V(xjl i ~ [) implies that a :;; V(xil i ~ 11)

for some finite 11 ~ 1. A lattice £ is algebraic if E is complete

and every element of E is a join of compact elements

If we take the S constructed in (a) from the algebra ~(, then

<S; ~ > is called the subalgebra lattice of ~L If e and I:}) are con

gruence relations of '\lI, then let e :5 I:}) mean that e is a refine

ment of I:}). Under this partial ordering, the set of all congruence

relations of ~r forms a lattice, called the congruence lattice of ~L

THEOREM The following four conditions on the lattice 2are 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

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, respec13

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 <s: be a group and lB an algebraic

lattice; then there exists an algebra ~ whose automorphism group

is isomorphic to (S; and whose congruence lattice is isomorphic to

lB, provided the lattice has more then one element.

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 by a finite

set of identities has to be recursive (Asser, Mostowski). No char- "

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 <a1"'" an"" > of

zeros and ones where an == 1 if n IE Sp(K), and an == 0 if n ISp(K). Let A be the set of spectra of equational classes and B the

set of spectra of equational classes defined by finite sets of iden

tities. A is known and B is not known. A and B are subsets of

2Ho, where 2 is the two element discrete topological space.

THEOREM. The closure of B in 2No is A.

~~

Let ~{ be quasi-finite if IP(n)(ml < NO for all n < (U.

Set cn == IP(n)(~l)1 and ~n == IP(n)( ~nl, where p(n)(~o is the set

of n-ary polynomials over ~{ depending on every variable. Let CK

and CK denote the set of all sequences <co' c I '··· >, <~O' ~l"'. >,respectively, which arise from a quasi-finite algebra ~{IE K.

SWierczkowski proved that if K is the class of all algebras,A N

then CK and CK are closed subsets of N 0, where N is the set of

all non-negative integers with the discrete topology. Does the same

conclusion hold whenever K is an axiomatic class or the class of

finite algebras? No characterization theorem is known for CK or CK •

6. Free algebras and identities. An algebra m: is called

free over a class K if ~l IE K and ~{ has a generating system H

such that for every )8 IE K and for every mapping ¢: H 4 B, there

exists a homomorphism 1> of ~l into )8 extending ¢. H is called

a basis of ~l. ~{is uniquely determined up to isomorphism by


m = IHI and K, and will be denoted by &K(m). If ~l is free over

K it is also free over HSP (K), that is, over the equational class gen

erated by K.

Let K be an equational class, and let Id(K) denote the iden

tities satisfied by all algebras in K. Then (G. Birkhoff) K ~ Id(K)

sets up a one-to-one correspondence between equational classes and

closed sets of identities. A set ~ of identities is closed if

(i) (xo = xo) ( ~;

(ii) (p = q) ( ~ implies (q = p) ( ~;

(iii) (p = q) (~, (q = r) ( ~ imply (p = r) ( ~;

(iv) if (Pi = q) ( ~, then (f/po"'" Pn -1) = fy(qo"'" qn -1))y Y

(2: for fy ( F; and

(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 &K1

«i)).

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 Koc K, Ko equational, then Ko contains only one element algebras)

and we call Id(K) equationally complete. Every set of identities that


can be satisfied by an algebra with more than one element can be

extended to an equationally complete one (A. Tarski) and there areIt

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)'

A. L. Foster's primal algebras8

(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 ~( can be

embedded in an algebra satisfying k.Free algebras are closely connected with another important

algebraic construction, namely the free product (R. Sikorski!6). The


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 . -J...I i f I) andI I 'f'I

whenever ~ f K and .1•. , i f I, are homomorphisms of \}l. into ~,'f', 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 'tYK(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.


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 elements 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'llis 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

jection functions e:, defined by e:(xo,. ..,xn_ 1) =: Xi'

Then

p(2l) ==

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,

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

also under equivalence (21 == <A; F> and 21 1 == <A; F1> are equiv

alent if p(n)( 2l) == p(n)( 2(1) for all n). Let us call 2l == <A,' F>

trivial if 21 is equivalent to <Ad~>, i.e., p(20 == 00, or IAI == 1.

The major unsolved problem of this field is, of course, the

characterization problem of the silt-tuple,

<IAI, g*, g, i, i*, p> •

Some general results in this direction are the following:

(0 IAI '? g* '? g '? i '? i* '? p if 2l is not trivial.

(Marczewski)

(n) i* == p or p + 1, and for i* '? 4 we have i* == p.

(Swierczkowski)

(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:

<X,y> Occurrences

1) <IA!,g*> l<n,m>;n 2: m::: O} - !<O,O>,<I,O>}

2) <!AI,g> l<n,m>;n::: m::: 01 -1<0,0>,<1,0>1

3) <IAI, i> l<n, m>; n 2: m 2: O} - 1<0,0>, <1, O>}

4) < IAI, i*> I<n,m>;n ::: m::: 01 -1<0,0>,<1,0>1

5) <IAI,p> I<n,m>;n> m::: l} U I<n,oo>;n::: l} U

U l<n,O>;n ?: 21 U I<n,-I>;n::: 21 u 1<2,2>1

6) <g*, g> I<n, m>; n ::: m ::: II U 1<0,0>1

7) <g*, i> I<n,m>;n 2: m::: 01

8) <g*, i*> I<n, m>; n 2: m 2: 01

9) <g*, p> l<n, m>; n > m 2: -11 U 1<2,2>1 U

U !<n, 00 >; n 2: II10) <g, i>. I<n, m>; n 2: m ::: 01

11) <g, i*> l<n,m>;n 2: m ::: 01

12) <g,p> l<n, m>; n > m 2: -11 U 1<2,2>1 U

U I<n,oo>;n > U13) <i, i*> I<n,m>;n ::: m ::: 01

14) <i, p> l<n,m>;n ::: m::: 01 U l<n,-I>;n::: O}

15) <i*, p> I<n,oo>;n 2: II U l<n,n>;n 2: 01 U

U !<3,2>,<2,I>,<I,O>,<O,-I>1

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 <no' nl'

...> all n j are 0 or 1).

Typical results are the following:

(i) Let ~l be a free algebra with p =: 2, i* =: 3. Then 21- 1

divideslAI·

(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 xn_ 1, then for some r ( p(n-l)( ~O, p(aO'·oo, an_I)

= q(ao,.oo, a 1) iff a. 1 = r(ao'··o, a 2)' for all ao"'" a 1 (n- 1- n- n-

A. (p = q can be distinguished by xn_ 1 means that there exist

bo'·'·' b l' b' 1 ( A with p(bo'··o, b 1) = q(bo"'" b 1) andn- n- n- n-

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 I is dependent, then a ([ao"." a 1] (the subalgebran n 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.

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 build1 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. <lXx. ''''tX. ) will usually de-10 1n _1

note a formula in which the x. have (or may have) free (unbounded)I j

occurrences.

It is intuitively clear what it means to say that <P(ao" .. , an_I)

holds in \I{, where ao' ... , an_I

£ A, and 2{ is an algebra of the

given type. For instance, if <P(xo) is

then cI>(a) holds for aU elements of a group <G;f> (where f is

the multiplication) and cI>(a) holds for the element a of the join

semilattice <A; f> if and only if a is the largest element.

A formal definition of "satisfaction" was given by A. Tarski 21.

A sentence <P has no free variables, hence it either holds or

does not hold for 2!. Thus a sentence expresses a property of the

algebra.

If ~ is a set of sentences, and all <P £.~ hold in 2{, then \I{ is

caUed a model of ~, or ~{ is called a ~-algebra.

First order languages are very restrictive: e.g., one cannot ex

press the property that 2{ is generated by a single element, since

we cannot express the phrase "there exists a polynomial p such

that ... " nor can one express the property that ~{ has no proper

subalgebra ("for all subsets B of A such that ... "). Thus many

of the algebraic properties encountered are not first order proper

ties. However, one can say much that is interesting and useful

about first order properties.

One of the most basic results is the following (J . .:t:,o!F);

If a sentence <P holds for aU ~r., i £ 1, then it holds for anyI

prime produce '){ of the \If.. In fact, it is enough to assume thatI

I i I <P holds for 2{.1 i L in order to assure that <P holds for• I

llL('){il i £ n.This means that the prime products of fields are again fields,

and the same holds for groups, rings, lattices, nudso on. An easy

corollary is the so-called "compactness theorem":

Let ~ be a set of sentences. If every finite ~o ~ ~ has a

model, then ~ also has a model.

It was A. I. Mal'cev, and later on, independently, A. Robinson

and L. Henkin, who pointed out that the compactness theorem has

important algebraic applications. Let us see a typical application.

An axiomatic class K is the class of all models of some set

of sentences. Let K be an axiomatic class; if all finitely gener

atedsubalgebras of ~( can be embedded in some algebra in K, then

~( can be embedded in some algebra in K. In fact, the same con

clusion holds if we assume only that K is closed under prime prod

ucts.

The same results 'hold if we consider algebras with relations,

which we will call structures. Then the compactness theorem can

be used to prove for instance that if all finitely generated subgroups

of the group <G ; . > can be ordered, then <G ; . > can be ordered.

The algebras (structures) ~1 and ~(2 are elementarily equiva

Ient if a sentence til holds in ~(1 iff til holds in ~(2'

A subalgebra ~(I of ~ is an elementary subalgebraof ~( if for

any formula tIl(xo' . n, xn _ l ) free at most in xo" no xn_ 1' if

tIl(aO' , an_I} holds in ~(I' for aO"'" atl

_ 1 ( AI' then

tIl(ao' , atl

_ l ) holds in ~. This very important concept is due to

A. Tarski21 •

A typical example is the following: Let ~( be the Boolean

algebra of allsuhsets of the set 1, and WIthe subalgebra consist

ing of the finite subsets of 1 and their complements. Then ~l: 1 is

an elementary subalgebra of ~l: .

The following result shows that there are always "small"

elementary subalgebras.


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

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.,

(xo)( 3xlh/t, (3x2)l/t, (x3)l/t, where l/t is the matrix. A di

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

®oV ... V ® l' where the ®, are identities or negationsn_ 1

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

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.Morel 23),

We cannot really expect that the converse statements of (a)-(e)

hold. For instance, <1>: (xO)(x1)(:3 x2)(x

Ox

l=x

1x

OV x

2.;, x

2) is

not universal, but it is preserved under the formation of subalgebras.

The reason is obvious: is equivalent to (xO)(x1)(x

Ox l >= x1x

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 <assume here.

Besides, a good survey on this (by Ju.L.Ersov, I.A.Lavrov, A.O.~ v

Taimanov, M. A. Taiclin) is available.

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 <G ; . , 1>, stating

that is associative, 1 is the identity and

(*)

Then the sub algebras 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 <L; V, /\, 0,1> with

the troublesome axiom

It is obvious that there is no "natural" way in which the comple-


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 2ft

is a

subalgebra of 9f, then ~£t is a I-subalgebra if all Ilinverses" in

2f of elements of Al

are also in At (the Hinverses" are those

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 o (At' at ( A, and (x2)t/r(aO' at' x2) in ~, then at ( At"

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'

A similar argument shows that a I.-homomorphism should be de

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


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 sayI I

that :£0 is better than :£ 1 if :£0 allows more subalgebras and homo-

morphisms. Is there always a best :£.? This gives a possibility ofI

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.

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 Kowas 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.


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. J6nsson).

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 classical result: 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 equational classes


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


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 '

~ s' Wl6

, andWlg have finite bases. Since these are lattices, it is

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

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>.

207

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 9CS

' since cI>())(s) holds for 2 if 2 is

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

isomorphic to 9JlS but 3 (2) has a sublattice isomorphic to ~ms)'

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

FOOTNOTES

209

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.


26. Varieties of Groups, Ergebnisse der Mathematik, Springer

Verlag, Berlin-West, 1967.

27. Duke Math. ]., 1966.

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