narrowness implies uniformity

Algebra Universalis, 15 (1982) 67-85 0001-5240/82/001067-19501.50+0.20/0 �9 1982 Birkh/iuser Verlag, Basel

Narrowness implies uniformity

RALPH MCKENZIE

Abstract . This paper is about varieties *//" of universal algebras which satisfy the following numerical condition on the spectrum: there are only finitely many prime integers p such that p is a divisor of the cardinality of some finite algebra in ~'. Such varieties are called narrow.

The variety (or equational class) generated by a class 5g of similar algebras is denoted by V(gg) = HSPSg. We define Pr (~) as the set of prime integers which divide the cardinality of a (some) finite member of ~/'. We call 5g narrow if Pr (5g) is finite. The key result proved here states that for any finite set X of finite algebras of the same type, the following are equivalent: (1) S P ~ is a narrow class. (2) V(gg) has uniform congruence relations. (3) SSg has uniform congruences and V(gg) has permuting congruences. (4) Pr (V(~)) = Pr (SSg).

A variety ~ is called directly representable if there is a finite set 5g of finite algebras sucla that = V(N) and such that all finite algebras in *t," belong to Pgg. An equivalent definition states that ~ is

finitely generated and, up to isomorphism, ~ has only finitely many finite directly indecomposable algebras. Directly representable varieties are narrow and hence congruence modular. The machinery of modular commutators is applied in this paper to derive the following results for any directly representable variety ~". Each finite, directly indecomposable algebra in ~ is either simple or abelian.

satisfies the commutator identity [x, y] = x �9 y �9 [1, 1] holding for congruences x and y over any member of ~ . The problem of characterizing finite algebras which generate directly representable varieties is reduced to a problem of ring theory on which there exists an extensive literature: to characterize those finite rings R with identity element for which the variety of all unitary left R-modules is directly representable. (In the terminology of [7], the condition is that R has finite representation type.) We show that the directly representable varieties of groups are precisely the finitely generated abeliar~varieties, and that a finite, subdirectly irreducible, ring generates a directly representable variety iff the ring is a field or a zero ring.

1. Prologue

This paper is in two parts. The first part (w and w contains an easy proof of the complete equivalence between narrowness and uniformity. This, the main

Research supported by U.S. National Science Foundation grant number MCS 77-22913. Presented by G. Gr~itzer. Received February 13, 1980. Accepted for publication in final form

January 12, 1981.

67

68 RALPH McKENZIE ALGEBRA UNIV.

contribution of the paper , should be accessible to anyone. The second half of the paper is of a different complexion, as we rely on some sophisticated machinery

developed in other papers. w is an a t tempt to relate the two parts, and the history of the concepts t reated in this paper will be briefly discussed there.

A theory of commutators that works in the setting of a completely arbitrary congruence modular ,variety was published just last year by J. Hagemann and C.

Herrmann. (In [13] and [14]. J. D. H. Smith's [21] was an important precursor.) Our work in w illustrates the power of this tool. This paper is the second of a s e r i e s - [9], [10], [3], and [ 1 7 ] - p u t t i n g the commuta tor to work.

2. Narrowness implies uniformity

T H E O R E M 2.1. I f SP(F) is narrow, where F is a finite algebra, then F has

un i form congruences .

Proof. Let 0 be any congruence of F and enumera te the cardinalities of the 0-blocks as a t . . . . . ak. (We assume that IF/Ol=k.) We have to show that a~ ak. For each integer n t> 1 define s~(6)= ~ " �9 �9 " . . . . . a l + a 2 + �9 + a k . (gt denotes the sequence (at . . . . . an).) s,(~i) is the cardinality of F, ~ S P ( F ) where the

universe of F, is

Fn = {or ~ "F : o'(i)Oo'(j) for all i, ] < n}.

Thus every prime divisor of s, (~i) is in the set Pr (SP(F) ) . We conclude the proof

of Theorem 2.1 by proving the following lemma.

L E M M A 2.2. I f a l , . . . , ak are positive integers a n d Pr ({sn(a): n/> 1}) is finite

then at = . " " = ak.

Proof. Suppose that the only primes p satisfying (3n)(p divides sn(ti)) are Pt . . . . . p,. Dividing all the a~ by d = g c d (a,, 1 <<- i <<. k ) and changing notation, we can assume that the greatest common divisor of all the a, is 1.

Claim. For p a prime, if we define b, = a ? with m = (p - 1)p k, then for all n >I 1,

pk+l does not divide s,(b).

To see this, observe that a~ - 1 - 0 or 1 (mod p) and, correspondingly, a ? - - 0 or 1 (mod pk+l). If we let a denote the number of distinct i ~ [1, k] such that p does

not divide ai, then l~<a since the ai have no common divisor other than 1.

Vol. 15, 1982 Narrowness implies uniformity 69

Further , for n >I 1,

s. (/~) = ~ a2n-= o~ (mod pk+l). i

Since 1 ~< a <~ k < pk+t, the claim follows.

F r o m the claim, it follows that if we put m = [~ 1.~ (Pi - 1)p~ then for n >/1, no

p~+l divides Smn(~i). Hence s,,,,,(gt)<~([-]l.,,pj) k since these are all its pr ime

divisors. Our sequence is bounded . This clearly implies that a l . . . . . ak = 1.

Remark. Afte r the first draft of this paper had circulated, several people

poin ted ou t that L e m m a 2.2 had been proved in 1915 by G. P61ya. It is a special

case of Satz I I on page 19 of P61ya's paper entitled "Ar i thmet i sche Eigenshaf ten der Reihenentwick lungen rat ionaler Funk t i onen" (J. reine und angewandte Math.

151 (1921), pp 1-31).

3. Locally finite uniiorm varieties

T H E O R E M 3.1. If S(2F) has uniform congruences, where F is a finite algebra, then F has permuting congruences.

Proof. Let 3" and 0 be arbi t rary congruences over F. We have to show that the relat ion products 3' ~ 0 and 0 o 3" are equal, where 3" o 0 ={(x, y):(3z)(XTZOy)}. Now A = 3 " o 0 is a subalgebra of 2F; define a congruence 0 0 " 0 t on it:

(x, y)00" Or(u, v) iff (x, u), (y, v) ~ 0. W e are regarding 00" 01 as a congruence on A. Since F e S(2F) - as the diagonal s u b a l g e b r a - both 0 on F and 00 �9 01 on A are

uniform. Let r = lx/ol be the uniform size of 0-classes in F, and s = t(x, y)/00" 0t[

the uni form size of 00" 01 classes in A. Of course, for x e F we have (x/O) x (x/O) ~_ A and hence s=l(x ,x) /Oo. O11=r 2. It follows that for any ( x , y ) ~ A , (x/O) x (y/0) ~ A, else I(x, y)/00" 0tl < r 2. This simply means that 0 o 3' o 0 o 0

3"o0, and a f o r t i o r i 0 O T _ 3 " o 0 . I t is well known that 0o3"_c3, o0 implies

0 o 3" = 3" o 0 for equivalence relations.

L E M M A 3.2. Suppose that D c_ Ao x A t projects onto Ao and A1 with projection kernels "0o, "or, where Ao and A t are finite algebras. Suppose further that "00 ~ "01 = "at ~ "00 and that one, at least, of Ao and A1 has uniform congruences. Then

IDI" ID/("0o +'00t = IAol" lad.

Proof. Lett ing zr~ deno te the pro jec t ion map of D onto A~, there are congru-

ences 01 =r on A~ such that

Ao/Oo ~ D/('0o + "01) ---- AI/01.

70 RALPH Mclr,.ENZIE ALGEBRA UNIV.

For the induced isomorphism cb:AolOo ~ A1/01, w e claim that

qb(x[Oo)=y[O~ iff ( x , y ) e D . (1)

Indeed qb(x[Oo) = y/01 ill there are (x, v), (u, y) ~ D with (x, v)(rio+ 711)(u, y). If this holds then, since rio permutes with "01, we have (x, V)~o(a, b)rlt(u, y) for some (a, b )~D; here (a, b )= (x, y) is forced by the definition of rio and "ql. It follows that

D=I,_J{C• CeAo/Oo}.

This union is a disjoint one. Suppose, uniform. Then (2) tells us that

(2)

without loss of generality, that Oo is

IDI = ~ Icb-l(y[OOl=lAtl " (IAol/n) y~A1

where n = IAo/Ool = ID/(no+ m){.

T H E O R E M 3.3. (Clark, Krauss [4]). If ~ is a locally finite variety such that all congruences on finite algebras in 7/" are uniform, then congruences on any algebra in

have the following properties: uniform, permutable, regular, coherent.

T H E O R E M 3.4. For any finite set 5~ of finite algebras (of the same type) the following conditions are equivalent.

(1) S P ~ is narrow. (2) V(~) has uniform congruences. (3) $5( has uniform congruences and V(~) has permuting congruences. (4) Pr (V(~)) = Pr (S~f). (5) Vr (SPN) = Pr ($5().

Remarks. The proof of Theorem 3.4 that follows is not self-contained because we use Theorem 3.3. If the second condition of Theorem 3.4 is replaced by "the finite members of V(5~) have uniform congruences," then a self-contained proof is easily given using Theorem 3.1, and using the fact that V(~) has permuting congruences if and only if its free algebra with three generators (which is a finite algebra) does. Theorem 3.1 is included here also for another reason: it constitutes a proof of the fact that a locally finite variety with uniform congruence relations has also permuting congruence relations, and this proof is much shorter than the ones to be found in the literature. It should be remarked that [19] has a somewhat different proof of Theorem 3.3 than the one found in [4].


Proof of Theorem 3.4. The implications ( 4 ) ~ ( 5 ) ~ ( 1 ) are obvious, and (2 )~ (3) is by Theorem 3.3. For (1)~(2) suppose that SPgg is narrow. Then for any A ~ S P ~ , SP(A)cSPSg is narrow, hence by Theorem 2.1, A has uniform congruences, if finite. It is well-known that every finite B ~ V(YD is in H(A) for some finite A ~ SP~, hence the congruences of B are uniform. (This property is preserved under taking homomorphic images, at least of finite algebras.) Now use Theorem 3.3.

For (3) ~ (4): Every finite member of SPSg is isomorphic to a subdirect product D = Ao x ' ' ' • where Aie SSg. Let P~ ( i< n) be the projection of D into Ao x . . . x Ai. By Lemma 3.2 and permutability, and by the uniformity of A,_~, IDI" IDIO.-d = Ie.-21" IA.-d where DlO,_t is a factor (i.e., homomorphic image) of D, A,-1 and P,-2- Applying the same step to P,-2 (in place of D) and multiplying the previous numerical equation by a constant, and then continuing on down, we arrive eventually at

n - - I n - - I

IDI " [~-] [D/O,I = [-~ [A,I (6)

where 01 , . . . , 0,_~ are certain congruences on D. Hence every prime divisor of IDI divides the cardinality of some A ~ SYL This shows that (5) holds, hence (1) and (2) as shown above. Then (4) follows, as every finite member of V(Sg) is the homomorphic image of a finite member of SPSg.

THEOREM 3.5. For any locally finite variety ~ the following are equivalent.

(1) ~ has uniform congruences. (2) For each finite F s ~ , SP(F) is narrow (3) For each finite Fe~ " and congruences ~/o, ~i on F, IFJ'VoO'/li divides

IFlvol " I F / y d .

Proof. That (1)r follows from 3.3, 3.4. Condition (3) obviously implies (2), and (1) implies (3) via 3.1, 3.2.

THEOREM 3.6. For a finite algebra F the following are equivalent.

(1) V(F) has uniform congruences. (2) S(F) has uniform congruences and V(F) has permuting congruences.

Note. Thus, to determine uniformity of the variety, check subalgebras of F for uniformity, and subalgebras of "F(n = IFI 3) for permutability.

Proof. By Theorem 3.4.


C O R O L L A R Y 3.7 (Quackenbush [19]). For a finite algebra F, where IF[ = k, the following are equivalent.

(1) Spec (SP(F)) ={k" :n ~to}. (2) Spec (V(F)) ={k" :n ~to}. (3) F is simple, has no proper subalgebras except 1-element algebras, and V(F)

has permuting congruences.

Proof. (The operator Spec is defined in the first paragraph of w Now (2)::>(3) is by Theorem 3.4.

(3)::> (1): This is a consequence of the well known and easily proved fact that a subdirect product of finitely many simple algebras in a permutable variety is isomorphic to a direct product of some of them.

(1)::> (2): We write an arbitrary finite A ~ V(F) as A =D[O where D ~ SP(F) is finite. We assume that (1) holds. By Theorem 3.4, the 0-classes all have the same cardinality, say c. Let d = IDI. Then the algebra 0 _ 2 D has c �9 d elements. By (1), d and c �9 d are powers of k, hence so is c. Then IAl= d/c is also a power of k.

Remark. The question as to how wide a finitely generated variety must be, if it is not narrow, has a satisfactory answer. (See paragraph 3 in w The question as to how wide a finitely generated variety must be, if it is not to have permuting congruence relations, is completely open.

4. Discussion

The spectrum of a class, Spec (~), is the set of cardinalities of finite members of ~ . The fine spectrum of a class, fx, is a function defined on the set of positive integers, f~(n) being the number of non-isomorphic n-element members of YC. G. Gr~itzer proved that the spectrum of a variety can be any monoid of positive integers. (I.e. any multiplicatively closed set containing the integer 1.) This is obviously not true for varieties generated by a finite algebra with a finite set of basic operations, because these give rise to a merely denumerable set of spectra. It is not even true for finite algebras with unrestricted operations, because there exists a certain infinite set of prime integers whose complement intersects, for each finite sequence ~i of integers such that not all ai are identical, the set Pr ({s,(a) : n ~ co}). It follows from w167 that the monoid generated by such a set of primes is not the spectrum of any finitely generated variety.

The problem of describing all fine spectra functions of varieties, or all spectra of finitely generated varieties, is completely open. A more realistic program for now would be to continue looking for relations between algebraic properties of a finitely generated variety ~ and conditions on Spec (~), or on ]'~-. (Incidentally,


finitely generated means that ~ = V(Sg) for a finite set of finite algebras N.) The results of w167 show the equivalence: ~ has uniform congruences r (~') is narrow. The third paragraph of w supplies examples of finite algebras F such that Spec (V(F)) is minimally un-narrow. V(F) does have permuting congruences in these examples, and it remains plausible that some condition on Spec (~) much weaker than narrowness can be shown to imply permutability, for finitely generated varieties.

W. Taylor's paper [23] is the standard source of information on varietal spectra. Taylor found all 2-element algebras F such that fv(r:~ gives just one algebra in each power 2 n, and no other algebras. Quackenbush's theorem on algebras of minimum spectrum (our Corollary 3.7) was an immediate descendent of this result of Taylor's. Independently of Quackenbush and during the same period, 1975-76, Clark-Krauss (represented by [4], [5], and [6]) and McKenzie ([15] and [16]) reached interesting conclusions about algebras slightly more general than those having minimum spectrum. This present paper owes its existence to all of the earlier work just mentioned, and in a sense culminates that work.

This paper establishes the following implications: direct representability (an algebraic concept, see w narrowness r uniformity ::~ permutability (an algebraic concept). The work done in 1975-76 contained many hints pointing toward these connections. That work was focused on some rather special varieties which were shown to have all of these properties. (The concept of a narrow variety was missing. The concept of a directly representable variety, introduced by Clark and Krauss, remained a mysterious locked door.)

In the next (more technical) section of the paper we study directly representable varieties using the commutator as our tool. Commutator theory is an analysis of 'centrality' in congruence modular varieties. J. D. H. Smith is credited with creating this theory (and publishing it in [21]) in the context of congruence permutable varieties. The extension to modular varieties followed almost im- mediately, although it was not at all trivial. (Hagemann and Herrmann accom- plished this in [13] and [14].) Perhaps there is historical interest in the fact that Smith was looking at some of the same problems motivating the previously quoted work of 1975-76, although from a rather broader perspective. In fact our [15]--[16] and his [21] (and we may mention also [12]) contain some common results arrived at independently.

5. Directly representable varieties

A variety is called directly representable if and only if it is finitely generated and has (up to isomorphism) only a finite set of finite directly indecomposable

7 4 RALPH MeKENZIE ALGEBRA UNIV.

members. We shall reduce the problem of characterizing finite algebras whose generated variety is directly representable to the special case in which V ( F ) is abelian. This special case will then be shown equivalent to a long standing open problem in the representation theory of finite rings,

Our principal tool will be the commutator. This is a binary operation that can be defined on the congruence lattice of any algebra A which belongs to a congruence modular variety. The commutator of two congruences is denoted [0, r it is a congruence on A. The first two propositions state properties of this operation which can be found in [13], [14] and [10] (which it is hoped will be available by the time you read this). Henceforth, all algebras mentioned are assumed to generate varieties with modular congruence lattices. The least and largest elements of the congruence lattice of A, Con A, are denoted by 0, 1 (or sometimes 0A, 1A), and the lattice operations in Con A are denoted by + and ..

P R O P O S I T I O N 5.1. For a, [3, 7~ (i ~ I) belonging to Con A we have:

(i) [a, 13] = [13, a] ~< a" 13. (ii) [a, V (7~, i e I)] = V ([a, %], i ~ I).

(iii) [1, 1] = 0 if[ A is polynomiaUy equivalent to a module over a ring. ( A is called abelian in this case.)

P R O P O S I T I O N 5.2. (i) For A c_ B, /3, "y~Con B, we have [ A ] /3, A I V] c-

[/3, v]. (ii) Let A = Bo x B1 with projections Zro, qrl and for ot ~ Con Bi put oti= ~rTl(ct)

Con A . Then for a, /3 ~ Con Bo and 7, 8 ~ Con B I we have

[ so- ~i, [30" 81] = [~, [3]0" [v, 811.

(iii) Let t~ : A --~ B be an onto homomorph i sm with ker ~ = a, and let K, X >I a in

Con A . Then ~-1([r ~(h)] )= [K, X ] + a. (This congruence is denoted by, [K, xL.)

Statements 5.1, (i) and (ii) and 5.2, (i) and (iii) can be found in [13]. Statement 5.1(iii) is proved in [14] and 5.2(ii) will be found in [10]. The commutator can actually be characterized in the following way: It is the largest binary operation on the congruence lattices of all algebras in a modular variety which satisfies Propositions 5.1 and 5.2.

By w the commutator is defined on any directly representable variety ~ . Consider the following identity:

x - [y, y]~<[x, y]. (c1)


If (C1) holds for all A ~ and for all x, y s Con A then we say that 3v satisfies (C1). We define another commutator identity:

[x, y ] = x . y �9 [1, 1]. (C2)

Any DR-var ie ty is residually finite hence by [9], Theorem 7,

L E M M A 5.3. Each directly representable variety satisfies (C1).

L E M M A 5.4. Every subdirectly irreducible algebra in a. directly representable variety is either abelian or simple.

Proof. Suppose not, say A ~ 3/" has monolith /3 (the smallest non-zero congruence), [1, 11/>13 (i.e. A is non-abelian), and 13 < 1 . We know that A is finite, and by (C1):

[/3, 1] =/3. (1)

Now employ once again the trick of w namely, for fixed n > 1 define

B ={f~ "A :f(i)/3f(j) for all i, j < n}.

Our goal is to show that B is directly indecomposable. As the size of such algebras (depending on n) is unbounded, we shall then have a contradiction.

For any 0 ~ C o n A and i<n, we define 0 , ~ C o n B to be {(f,g):f(i)Og(i)}. Clearly/30 =/31 . . . . . /3,-1 (from the definition of B), call this congruence/3. In this notation 0~ is the kernel of the i-th projection from B onto A, and we define 0' =/ki,,i 0j. It's easily verified that 0~ + 0~ = 0, + 0 m- (j p i) = V i<, 0' =/3. Suppose that B is directly decomposable and we shall argue to a contradiction. We have congruences on B satisfying

qJ+X = 1, tO.X=0, @>0, X > 0 . (2)

We claim that for i < n:

([Of, X] = 0~ and [Of, ~0] = 0) (3)

or else

([Of, ~b] = O' and [O', X] = 0).


First, by 5.2(iii) and (1),

/3 = [/3, 1]o, = 0~ + [0, + 0~, r + X] = 0i + [0;, 0] + [0" X]. (3')

Moreover/3 =/3~ covers 0i (since/3 covers 0 in Con A), and 0' covers 0 since/3/0i transposes down to 0'/0. We have [0', to] <~ Of. to and [0', X] <~ 0f. X and to- X = 0, hence [Of, tO], [Of, X] E 0'/0 are disjoint. Therefore one at least of these elements in the two element interval 0f/0 is 0, and the other must be Of, by formula (3'). Hence (3) is proved.

Now put S x = {i : [Of, X] = Of} and let S, be defined similarly. By (3), S x U S, = {0 . . . . . n - 1 } = n and SxAS ,=~ .

If S,:P 0 then X~</3; if Sx7 ~ O then to---</3. (4)

Indeed, let i~ S, and X~/3-Then XN 01, so 0i + X ~> fi =/31 (remember that/3 is the monolith of A-~B/O~). By modularity,

fi =O, +[J'x =O, +(V O; + V OL)" x S, S,~

=o,§ = 0~ + V 0~ = 0i

Sr

since Vs. 0;~<to- This is a contradiction, proving (4). Now since/3 < 1 on B and since to+X = 1, it follows from (4) that, say, S• = ;~

and hence S, = n. Then X ~</3 by (4), and /3 = V~ 01 ~< @, so x ~< to contradicting (2).

T H E O R E M 5.5. Each directly representable variety satisfies (C2).

Proof. Suppose not. For some A ~ and 0, f f ~ C o n A we have [0, to]< 0-O .[1, 1]. There is a congruence o~ such that A/a is subdirectly irreducible, [0, O]<~a, and 0.@.[1,1]No~ (Birkhoff's theorem). Now use 5.4. If A/ot is abelian then [1, 1], =o~ (see 5.2(iii)), i.e. [1, 1]~<a, which is false. Thus a is a co-atom of Con A (A/oL is simple). Then 1 = 0 + a = r since a dominates neither. Hence

[1, 1]= [0, $]+[0 , a]+Ea, $ ]+ [a , a]~<a

by 5.1(ii), a contradiction.


C O R O L L A R Y 5.6. The directly representable varieties of groups are exactly the finitely generated abelian varieties. The directly representable varieties of rings are exactly those generated by a finite set of ~nite fields (and a zero ring if we are talking about rings without unit).

Proof. The modular commutator becomes the ordinary commutator on groups. A variety of groups satisfying (C2) has no solvable non-abelian group. ((C2) implies that [1, 1]=[[1, 1], [1, 1]].) But every minimal finite non-abelian group G (i.e., every proper subgroup of G is abelian) is solvable. For this fact, see [24].

On rings the modular commutator of two ideals J and K is the ideal generated by all products jk and kj where ] e L k ~ K . A ring R is 'abelian' iff R2={0}. Suppose that R is a subdirectly irreducible ring in a directly representable variety of rings. By Lemma 5.4, if R is not a zero ring then it is a finite simple ring and R z 7~{0}. It is known from the theory of the Jacobson radical that R must be isomorphic to M, (F), the ring of all n x n matrices over a finite field F. Here if n >I 2 then the upper triangular matrices form a ring S _c M, (F) which (it is easy to see) is subdirectly irreducible and neither simple nor 'abelian.' Thus n = 1, and R is a finite field.

The converse statement, that any variety generated by finitely many finite fields and a finite zero-ring is directly representable, follows by Theorem 5.11.

DEFINITION 5.7. (1) The center of A is the largest congruence 0 satisfying [1, 0] = 0.

(2) A n element c in a lattice L is called neutral if for all x, y ~ L the sublanice generated by c, x and y is distributive.

L E M M A 5.8. (1) A n element c of a modular lattice L is neutral if and only if one (in fact both) of the following conditions hold: c(x +y) = cx +cy for all x and y in L; ( c + x ) ( c + y ) = c + x y for all x and y in L.

(2) I f Con A satisfies (C1) then [1, 1] is a neutral element in Con A, in fact [1, 1]- 0 --- [1, 0] for all congruences O.

Proof. (1) is from [11]. Now suppose that (C1) holds in Con A. Then [1, 1]. 0 ~<[1, 0]~<[1, 1]-0 so equality holds. Hence [1, 1] is neutral, by statement (1) and 5.1(ii).

T H E O R E M 5.9. Each finite algebra A in a directly representable variety

satisfies

(1) [1, 1] and ~ (the center of A ) are complements and neutral elements in Con A. Thus A = A/[1, 1] x A/ ~ and this carries with it an isomorphism of Con A with the lattice (1/[1, 1])x (1/~).

7 8 RALPH McKENZIE ALGEBRA UNIV.

(2) ([1, 1], ~) is the unique pair of complements (a,/3) in Con A such that A / o ~ [ 1 , 1]=0 and A/r 1]= 1.

(3) A/~--- B1 x . �9 �9 x Bk, a product of simple non-abelian algebras, and this decomposition is unique, in fact 1/~ ~- k 2.

COROLLARY 5.10. Every directly indecomposable finite algebra in a directly representable variety is either abelian or simple.

Proof of Theorem 5.9. Let A be finite in the DR-variety ~. We apply Birkhoff's theorem and obtain an irredundant (minimal) subdirect representation

trt

0 =/k 0~ (with A/O~ subdirectly irreducible). 1

(4)

By Lemma 5.4 we can assume that

0i+[1, 1]= 1=[1, 1]o, and 0i is a co-atom for l<<-i<~k

(A/O~ is simple and non-abelian; here k = 0 is possible);

[1, 1]~<0~ for k + l ~ i < - m (A/O, is abelian).

(5)

Let ~ = / ~ 0i, ,k =/k~'+l 0i. By permutability, irredundance, and maximality of 0~ (1 <<- i <~ k), A / ~ = ~ ~ A/Oi. Then A / ~ ~[1, 1] = 1 follows from Proposition 5.2. Translating this into Con A, we have tz+[1, 1]= 1. But we also have [1, 1]~<k and k �9 tz = 0. Adding these facts together and blending with modularity gives us ~t =[1, 1], [1, 1 ] + ~ = 1, [1, 1 ] . ~ = 0 . Thus

A ~-AI[1, 1 ] x A / O l x " " �9 x A I G . (6)

We next show that t~ = ~ (the center). Certainly, [1,/~] = [1, 1]-~ = 0. Con- versely, if 3,> fz then 31.[1, 1]>0 (modularity), hence [1, ~/]>0 (by 5.8(2)).

The above argument also shows that (a,/3) = ([1, 1], ~) is the unique pair of complementary direct factor relations satisfying [1, 1]~<a, [1, 1]+/3=1. It is easily verified that since [1, 1] is neutral (5.8)(2) its complement ~ is (unique and) neutral.

The only thing remaining is to prove the uniqueness in (3). This certainly follows from the assertion that l/~----k2. This assertion is a consequence of the fact that the interval 1/~ is isomorphic (projective) to [1, 1]/0 which is distributive by (C2).

THEOREM 5.11. Let 5( be an arbitrary finite set of finite algebras. The following three conditions are jointly necessary and sufficient in order for V ( ~ ) to be directly representable.

VoI. 15,

(1)

(2)

(3)

1982 Narrowness implies uniformity 79

There is a term t(x, y, z) in the language of ~g such that the identities t(x, x, y) ~ t(y, x, x) ~ y hold in each member of 5g (i.e. V(N) has permuting congruences). Every member of SSg is (isomorphic to) a direct product of simple algebras and abelian algebras. The variety generated by the set of abelian direct factors of subalgebras of members of 5g is directly representable.

Note. Given that (1) and (2) hold, the variety mentioned in (3) is the class of all abelian members of V(5~).

To prove the theorem we need

L E M M A 5.12. Suppose that A = Box . �9 �9 x B,,-1 where each Bi is simple and non-abelian. Then (i) [4', X] = O" X holds for any two congruences O, X on A, and (ii) Con A ~ m 2.

Note. This lemma is a special case of [13], Corollary 4.2. There are no unstated hypotheses except that V(A) is assumed to be congruence modular. In particular, A need not be finite. Also, the assumption that A = B o x . . . x B , , _ t can be weakened (as in [13]) to: A _~Bo x . . - xBm_~ is an irredundant subdirect product.

Pro@ By Proposition 5.2(ii) condition 5.12(i) will hold for congruences O and X on A which are intersections of projection congruences 0j (] < m). In particular A satisfies [1, 1]= 1. We have to show that every congruence 0 of A is an intersection of a subset of the projection congruences. For K ___ m denote OK = A(Ok, k ~ K ) and put J={j:Oc_O~}. Now whenever 0 + ~ = 1 = 0 + / 3 then 0 + a . /3 = 1, because

1 =[1 , 1] = [0 + a , 0 +/3]~< 0 +[a,/3]~< 0 +o~ ./3.

Thus 0 + 0m-j = 1 (by maximality of each Ok). Then

0~ = 0j" (0 + 0,,_j) = 0 + 0j" 0,~_j = 0 (by modularity).

Proof of Theorem 5.11. The necessity of the conditions comes from 3.4, 5.10. For sufficiency, let 5o denote the class of simple non-abelian direct factors of subalgebras of members of :g and let sr denote the variety generated by the abelian direct factors of subalgebras of members of ~. ,50 has only finitely many non-isomorphic members and sr is directly representable, by assumption (3). It will suffice to prove that every finite member of V(Sg) is in P(Se tA sr


First we consider a finite algebra F in SP~. By (2) F is isomorphic to a subdirect product.

D c A x B l x . . . x B k , where A ~ t and {B1 . . . . . Bk}_cgo. (4)

Repeating without change the first part of the proof of 5.9 we find that we can assume that D/IX ~-B~ x . �9 �9 x Bk, that h ' t~ = 0, and that h >/[1, 1] where h is the kernel of D ~ A. By 5.12, [1, 1]+/~ =[1, 1]~ = t, hence ~ =[1, 1] by modularity, and (4) is equality, not just inclusion.

Now let 0 be any congruence on D as above. By 5.12(i), /x+[1 ,0]= ~+[1 , u .+0]=[1 , ~+0]~ = u.+0. Hence [1, 1]-(/x + 0) = p..[1, 1]+[1, 0]=[1, 0] (modularity and [1, 0]<[1, 1]) . Thus 0= 0+[1, 0]= 0+[1, 1](IX + 0) = ([1,1]+O)(Ix+O). This shows that D / O = A ' x B ', A 'es~, B ' ~ H ( B I x . . "xBk). By 5.12(ii), B ' - I--1 (B~, i e I ) for some I___{1,..., k}.

Since every finite member of V(~') is isomorphic to some D/O as above, we are done.

THEOREM 5.13. Each finitely generated, semi-simple, congruence permutable variety is directly representable.

Proof. Let ~ satisfy the hypotheses. Let 9 ~ be the class of simple (or, for this ~ , subdirectty irreducible) members of ~. Of course ~ is generated by a single finite algebra A having, say, n elements. From a theorem proved by J. B. Nation and W. Taylor for the permutable case and generalized to the modular case in [9] (as Corollary 17), every member of 9 0 has at most n-elements. Thus they are all factors of the ~ algebra with n generators, a finite algebra. Consequently 5~ has only finitely many members, up to isomorphism. For the same reason as in the second sentence of the paragraph following statement (5) in the proof of Theorem 5.9, each finite member of ~ is isomorphic to the direct product of some members of 5~

Remark. The preceding theorem becomes false if the assumption of permutability is replaced by modularity, or even by distributivity; consider the variety of distributive lattices. The referee of this paper contributed the remark that for V(~) to be semi-simple, where 5g is a finite set of finite algebras and the generated variety is assumed to be congruence permutable, it is sufficient that every member of S ~ is isomorphic to a direct product of simple algebras. Thus if the second condition in Theorem 5.11 is strengthened in this way, then the third condition becomes redundant. Another way to put this: Any abelian modular variety generated by finitely many finite simple algebras is directly representable,


and semi-simple. Now the ring R of such a variety (described in the proof of Theorem 5.14 below) is a finite semi-simple ring with identity (the simple generating algebras give rise to irreducible R-modules which, taken together, are faithful for R) ; hence the direct representability and semi-simplicity can be deduced along the lines of the argument for 5.14 from the fact that every unitary module over a finite semi-simple ring with identity is completely reducible as the direct sum of irreducible modules. (An irreducible module is, by definition, a module which is a simple algebra and has non-zero multiplication.) The referee also indicated a direct universal algebraic proof of his remark. These remarks, and also Theorem 5.13, are probably known but we have not found them in the literature.

We come at last to our final theorem.

T H E O R E M 5.14. The problem of effectively determining whether or not a finite set of finite algebras (of finite similarity type) generates a directly representable variety reduces (and is equivalent) to the problem of effectively determining, given a finite ring R with identity, whether or not the variety RM of unitary left R-modules is directly representable.

Proof. It is clear that the second problem is a special case of the first. (RM is generated, as a variety, by the ring R regarded as a left R-module.)

On the other hand, of the three conditions for direct representability of V(~) given in Theorem 5.11, one can effectively (although it may be an impractically long and tedious exercise) check whether (1) and (2) hold by known procedures. Condition (3) is the sticker. We can at least compute the s~-free algebra F~(2) freely generated by two elements x, y, where ~ is the abelian variety generated by the abelian direct factors of subalgebras of members of 5g.

A definitive basic treatment of abelian varieties has not yet been written, but the following information can be gleaned from [14], or from [10] if it is available. Firstly, there is a term S(x, y, z) in the language of s~ (which can be found by calculating F~(3)) such that for any A ~ d and any element 0~ A there is an abelian group (A, + , - , 0) for which x - y + z = S(x, y, z) holds for all x, y, z ~ A (in particular, x + y = S(x, O, y)), and such that for any term "r(xl . . . . . xj) in the language of ~/ there are endomorphisms "~//1 . . . . . ~/j of (A, + ) and an element c ~ A such that

i �9 ( y l . . . . . = v , ( y , ) + c

1

holds for all Yl . . . . . Yi ~ A. (All of these facts are expressible by identities that must hold throughout ~/.)

82 RALPH McKENZIIE ALGEBRA UNIV.

Secondly, there is a ring with identity element R, which can be constructed from F~(2) as follows. The universe of R is the set of 'words' a = a(x, y)~ F~(2) such that the identity a (x, x) ~ x holds in ~ . The operations of R are obtained as follows: a ' / 3 ------ ol(/3(x, y), y), ct + [J - S(a(x, y), y, 13(x, y)). R is an associative ring with identity e lement x and zero element y.

Thirdly, if we go back to an arbitrary A a ~ / and choose 0 ~ A and look at the abelian group (A, +, - , O) as in the paragraph just above the last, we can define, for x ~ A and o ~ R , o~. x=a(x,O) as computed in A. This gives us a unitary R-module M(A) . Up to isomorphism of R-modules, M(A) is independent of the choice of 0.

Fourthly, the correspondence of A with M(A) maps ~t onto RM. Further, A and M(A) have precisely the same polynomial functions (term functions with constants replacing some variables), hence the same congruences and the same direct decompositions. Hence ~ and RM are both directly representable, or both not. (For each of these varieties, direct representability is equivalent to the existence of a finite bound on the size of directly indecomposable algebras.)

This concludes our proof.

Remark 5.15. Concerning the problem of ring theory to which 5.14 reduces us, the reader may consult [7] and [8] and their bibliographies, but we warned that these make heavy reading. I am informed that a ring R is said to be of finite representation type if and only if RM has only finitely many non-isomorphic directly indecomposable modules of finite dimension. It seems that the characterization of finite rings of finite representation type is known for finite rings whose radical squares to zero, and for very few others.

6. Epilogue: some problems and remarks.

I am very much indebted to Stanley Burris for stimulating correspondence during the months just prior to these discoveries. After finding Theorem 2.1, I came to a belief that something like Lemma 5.4 might be provable upon recalling Burris' papers [1 ,2] in which he showed that the necessary and sufficient condition for a congruence distributive variety to be directly representable is that it be finitely generated, semi-simple, and have permuting congruences. (If ~ = V(~f) satisfies Theorem 5.11(1-3) then "V is congruence distributive iff HS(~) has only trivial abelian algebras.)

The members of the Hawaii seminar of December 1979, Day, Freese, Lampe, and Nation, and later the anonymous referee, supplied much constructive criti- cism which (I hope) improved this manuscript.

Let me conclude with a few technical points and unresolved problems.


1. The proof of Theorem 3.4 establishes as well that the following are equivalent, for a finite set 5~ of finite algebras. Pfi~ (the class of subdirect products of members of Y~) is narrow; the finite members of P ~ have uniform congru-

ences; Pr ( P ~ ) = Pr (~). 2. In [20] it has been shown implicitly that condition (1) of Corollary 3.7 can

be weakened to: Spec (S(~F)) c_ {1 . . . . . k"} where n = m a x (4, k). 3. This paragraph answers in the negative a conjecture of R. Quackenbush.

Let 6 = (a~ . . . . . ak) denote any non-decreasing finite sequence of positive integers with al<ak, and for n~>0 put s , ( ~ ) = a ~ + . . - + a ~ . Write ~r(~) for the multiplicative closure of {1}U{sn(fi):n i>0}. We proved that any finite algebra F satisfies: Pr (V(F)) is infinite r Spec (V(F)) _ 7r(~) for some such sequence 6. It is interesting to note that ~(~) is a spectrum, in fact ~'(6) = Spec (V(F(~))) where the finite algebra F(~) is constructed as follows. The universe is a set of sl(a) elements and this set is partitioned by an equivalence relation R whose equivalence classes have cardinalities al . . . . . ak. The term functions of F(t~) are exactly all operations on the universe which preserve the relation R. (We can find finitely many operations to generate these, and so make F(t~) an algebra of finite type.)

F(ti) is, of course, one of the pre-primal algebras in class 3 of the Rosenberg classification (see [20]). Spec (V(F(~) ) )= ~-(~) is not narrow, and also not very wide. For instance, if ~ = (1, 9) then Pr (~-(t~)) contains no primes congruent to 3

(mod 4). Question: If F is finite and Spec (V(F))= ~'(1, 9) must V(F) be congruence

permutable? 4. The spectrum of any directly representable variety is a finitely generated

monoid. The spectrum of a variety generated by a finite non-abelian group is a finitely generated monoid, although this variety is not directly representable. A six element algebra F(3, 3) constructed as in paragraph 3 above generates a variety whose spectrum, namely -rr(3, 3), is narrow and not finitely generated.

5. The theorems about algebras of minimum spectrum furnished some first examples of non-trivial conditions that a function must satisfy in order to be a fine spectrum function of a variety. The results in this paper furnish some more. For example, let ~ be any variety such that Spec (~) _ {6" : n ~ ~0}, and suppose that b = f~-(6) is a finite positive integer. Then f~-(6") is at least as big as the number of partitions of n into b parts. This follows from the claim that if B1 . . . . . Bb are all the non-isomorphic members of ~ of cardinality 6, then any two products l-I~=~ B (~) with each B")~{B~ . . . . . Bb} are isomorphic iff they have the same number of copies of each Bi; in fact D=II~=~ B (~ has the unique factorization property. The claim is established thusly: V(B~ . . . . . Bb) is congruence modular (and permuting) by Theorem 3.4. This variety has no non-trivial finite abelian algebras, because every simple abelian algebra has cardinality a power of a prime.


Hence each Bti)~[1, 1] = 1 (Bti)/[1, 1] is abelian). The B t~) are simple. Hence by Lemma 5.12, Con D =n2. (Actually, it can be proved that V(B~ . . . . . Bb) is a discriminator variety, see Burris [2].)

6. Taylor constructed weak Mal'cev conditions for congruence uniformity of finitely generated varieties. Can our results be used to put his conditions into a more perspicuous form? Or vice versa?

7. What are the conditions on a DR-variety in order that it have no infinite directly indecomposable members?

8. Suppose that ~ is directly representable. It can be proved that statement (1) of Theorem 5.9 holds for every A ~ itt ~ is the join of an abelian variety and a congruence distributive variety iff for each non-abelian simple algebra S in ~ , every subalgebra of S satisfies [1, 1]= 1. Modular varieties ~ =~ 1 v ~ 2 where ~x is distributive and ~ abelian are studied in [13], and in [10] where it is shown that this formula actually implies that ~ = ~ @~2, the varietal product of ~ and 0//'2 in Taylor's sense.

9. Let ~ be any DR-variety and let n be the maximum cardinality of the non-abelian simple algebras in ~ . Following Burris [2], we look at F~r(n + 2), the ~-f ree algebra generated by, say, x, y, z0 . . . . . z~_ 1. By Theorem 5.9 there is a term D = D(x, y, Zo . . . . . z , -O in our language such that (identifying D with the element of F r ( n + 2 ) it represents), (x ,D)~[1 , 1] and ( y , D ) ~ (the central congruence of F~.(n + 2)). One can prove that D has the following properties. (i) The identity D(y, y, ~ ) ~ y holds in o//. and also certain identities expressing that D(x, y, 2 ) - y (modulo the center) in F,~(n +2). (ii) For any a, b ~ A ~'F', a and b are congruent modulo the center of A iff A~(V~:)(D(a, b, ~)= a), and they are congruent modulo [1,1] iff A~(3~.) (D(a,b ,~ . )=b) . (iii) If a, b, Co . . . . . c,_~ generate A ~ ~ then a -- D(a, b, ~) (modulo [1, 1])--- b (modulo the center of A).

REFERENCES

[1] S. BURRIS, A note on directly indecomposable algebras, Preprint (1979). [2] S. BURaIS, Arithmetical varieties and Boolean product representations, Preprint (1978). [3] S. BURP, S and R. MCKENZm, Decidability and Boolean representations, Preprint (1980). [4] D. CLARK and P. KaAuss, Para primal algebras, Alg. Univ. 6 (1976), 165-192. [5] D. CLARK and P. KaAUSS, Varieties generated by para primal algebras, Alg. Univ. 7 (1977),

93-114. [6] D. CLARK and P. KRAUSS, Plain para primal algebras, Alg. Univ. (to appear). [7] V. DLAB and C. M. RINGEL, On algebras of finite representation type, J. Algebra 33 (1975),

306-394. [8] V. DLAB and C. M. RtNGEL, Indecomposable representations of graphs and algebras, Memoirs

A.M.S. 6, number 173 (1976), 57 pages. [9] R. FREF.SE and R. MCKENZm, Residually small varieties with modular congruence lattices, Trans.

Amer. Math. Soc. 264 (1981), 419-430. [10] R. FREESE and R. MCKENZtE, The commutator, an overview, Preprint (1981).


[11] G. GRATZER, A characterization of neutral elements in lattices, Publications of the Math. Institute of the Hungarian Academy of Sciences, 7 (1962), 191-192.

[12] H. P. GUMM, Algebras in permutable varieties: geometrical properties of affine algebras, Alg. Univ. 9 (1979), 8-34.

[13] J. HAGEMANN and C. HERRMANN, A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity, Arch. Math. (Basel) 32 (1979), 234-245.

[14] C. HERRMANN, Affine algebras in congruence modular varieties, Acta Sci. Math. (Szeged), 41 (1979), 119-125.

[15] R. McKENzIE, On minimal, locally finite varieties with permuting congruence relations (unpublished).

[16] R. MCKENZm, Para primal varieties: A study of finite aziomatizability and definable principal congruences in locally finite varieties, Alg. Univ. 8 (1978), 336-348.

[17] R. MCKENZIE, Residually small varieties of K-algebras, Alg. Univ. 14 (1982), 181-196. [18] R. QUACKENBUSH, Algebras with small fine spectrum (unpublished). [19] R. QUACKENBUSH, Algebras with minimal spectrum, Alg. Univ. I0 (1980), 117-129. [20] R. QUACKENaUSI-t, A new proof of Rosenberg's primal algebra characterization theorem, Preprint

(1980). [21] J. D. H. SMrrn, Malcev Varieties, Springer Lecture Notes 554 (Berlin, 1976). [22] W. TAYLOR, Uniformity of congruences, Alg. Univ. 4 (1974), 342-360. [23] W. TAYLOR, The fine spectrum of a variety, Alg. Univ. 5 (1975), 263-303. [24] W. R. SCOTT, Group Theory, Prentice-Hall, New Jersey (1974).

University of California Berkeley, California

U.S.A.

narrowness implies uniformity

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