Semigroup Forum Vol. 55 (1997) 299{308c 1997 Springer-Verlag New York Inc.

RESEARCH ARTICLE

Congruence compact acts

Peeter Normak

Communicated by L.M�arki

Abstract

In this article we discuss some problems related with congruence compactness ofacts. It is proved that every monoid S which is congruence compact as a left S-act isquasiperiodic and that every congruence compact act over a commutative monoidis f-equationally compact. We also present an example showing that congruencecompact acts over arbitrary monoids are not necessarily even weakly 1-equationallycompact. Congruence compact acts over monogenic monoids and commutative groupsare described.

1. Preliminaries

Under the usual �niteness conditions there are two universal ones which can beconsidered in an arbitrary category of algebras, namely the maximum and minimumconditions for congruences. Other maximum and minimum conditions (for example,for ideals of semigroups or for other ordered structures) are more or less speci�c for theclass of algebras considered. In many cases, the maximum and minimum conditionsfor congruences are too strong and therefore a generalization of these conditionsmay be necessary. In the theory of rings and modules, the study of congruencecompactness as a generalization of minimum condition for congruences (as well asa generalization of topological compactness), was started more than forty years agoalready. The main contribution to this subject in the last years is probably due toP.N.Anh, who proved, for example, that the commutative congruence compact ringsare exactly those which admit Morita duality ([1]).

In ring- and module theory the notion \linear compactness" is used instead of\congruence compactness". The adjective \linear" is fully justi�ed there but cannotbe explained in a number of other classes of algebras. This is why for general algebras,the term \congruence compactness" will be proposed in the present article.

It should be mentioned here that \congruence compact" general algebrasand acts (under the name \linearly compact") were introduced in [5] and in [8],respectively. In [8], the relations between congruence compactness on the one handand equational compactness and Morita duality on the other hand were considered.In this paper we start studying the structure of congruence compact acts and monoids(as acts over themselves) for which purpose we devote the second section; the thirdsection deals with congruence compact acts over monogenic monoids and the �nalsection with congruence compact acts over groups.

Besides of the description of the structure of congruence compact acts andmonoids, there is another question we try to answer in several particular classes ofacts, namely: are congruence compact acts (f-)equationally compact?

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Let S be a monoid (with identity 1). A left S-act SA is a set A on whichS acts from the left, that is, 1�a = a and (s�t) � a = s�(t�a) for all a2A; s; t2S . Aset F = fCigI is a �lter base if for every �nite subset fCiji = 1; : : : ; ng of the setfCigI there exists an element C 2 F such that C � \n

i=1Ci . A congruence on aleft S-act SA is a Rees congruence if there is at most one congruence class containingmore than one element and this class is a subact of SA:

A monoid S is said to be:

- a left(right) Rees monoid if every left(right) congruence on S is a Reescongruence;

- right reversible if any of its two left ideals intersect.

An S-act SA is said to be:

- equationally compact if every �nitely solvable system of equations of A issolvable;

- 1-equationally compact (f-equationally compact) if every �nitely solvable sys-tem of equations of SA all containing one and the same variable (containing altogethera �nite number of variables, respectively) is solvable;

- congruence compact in the discrete topology or simply congruence compactif every �nitely solvable system of congruences (x; ai) 2 �i , i 2 I; is solvable. Thismeans that an act is congruence compact if and only if every �lter base consisting ofclasses of congruences, has a nonempty intersection;

- cyclic if it is generated by a single element;

- connected if from SA = SA1

`SA2 it follows that either SA1 = � or

SA2 = �. SA is obviously connected if and only if for any two elements a; b 2

SA there exist elements x1; : : : ; xn 2 SA and s1; : : : ; sn; t1 : : : ; tn 2 S such thata = s1x1; t1x1 = s2x2; : : : ; tn�1xn�1 = snxn; tnxn = b ;

- simple if it does not have proper subacts.

By < H > we denote the smallest congruence relation on an S-act SA

containing H � A�A . Note that coproducts in the category of all S-acts are disjointunions. The set of natural numbers (with zero) is denoted by N (N0 , respectively)and the set of integers by Z.

We proceed now with some elementary lemmas.

Lemma 1.1. Every subact and every factor act of a congruence compact S-act iscongruence compact.

Lemma 1.2. If SA satis�es the minimum condition on congruences then SA iscongruence compact.

Remark that the converse of Lemma 1.2 is not true. Take for example S =(� [ (g; 1]; �), a real interval, where 0 < g < 1 and x � y = � , xy � g . S

is a Rees monoid by [6, Satz 3.11 and 3.15] and hence congruence compact by [8,Proposition 5.2]. Hence congruence compactness does not imply even minimumcondition for subacts (and the less for congruence classes). The same example showsthat commutative congruence compact monoids are not necessarily cogenerators inthe Morita category SC = HSL(S), i.e. in the full subcategory of the category of all

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left S-acts containing S and closed under taking subacts and factor acts, because S

does not contain the (subdirectly irreducible) two-element S-act (� [ 1; �).

2. About the structure

In the present section we prove some basic properties of congruence compact S-actsas well as prove that every congruence compact monoid is quasiperiodic.

Let us start with some lemmas:

Lemma 2.1. If SA =`

I Ai is congruence compact, then jIj <1.

Proof. Assume jIj = 1 . For every �nite subset J 2 I consider the Reescongruence �J on SA : x�Jy , x = y 2 Ai; i 2 J; or x 2 Ak and y 2 Al withk; l 62 J . Then the nontrivial congruence classes of all �J form a �lter base whichhas an empty intersection.

Lemma 2.2. A coproduct SA =`n1 Ai is congruence compact if and only if every

SAi; i = 1; : : : ; n; is congruence compact.

Proof. Necessity is clear. Su�ciency. Prove by induction on n. For n = 1 thelemma is trivial, so assume the lemma is true for all n < k . Let n = k and letfCigI be a �lter base of congruence classes on SA . If fCi \ A1gI were a �lter basewe would have \ICi � \I(Ci \ A1) 6= �. Assume that fCi \ A1gI is not a �lterbase, i.e. there exist a �nite number of congruence classes, say, C1; � � � ; Cm , such that\m1 (Ci \ A1) = �. Next we will show that then fCi \ (

`nj=2Aj)gI is a �lter base.

Assume fCi \ (`m

j=2Aj)gI is not a �lter base. Then there exist a �nite number ofcongruence classes, say, C 0

1; : : : ; C0

l ; such that \l1(C

0

i \ (`n

j=2Aj)) = �. Denote theindex set of congruence classes from the set fCi; i = 1; : : : ;mg [ fC 0

i; i = 1; : : : ; lg byJ . Then we have � = (\m

1 (Ci\A1))[(\l1(C

0

i\(`n

j=2Aj))) � (\J(Ci\A1))[(\J(Ci\(`n

j=2Aj))) = (A1 \ (\JCi)) [ ((`n

j=2Aj) \ (\JCi)) = (\JCi) \ (A1 [`n

j=2Aj) =(\JCi)\A = \JCi , a contradiction. Hence fCi \ (

`nj=2Aj)g is a �lter base and has

a nonempty intersection, by induction.

Proposition 2.3. The following properties of a monoid S are equivalent:

1) all �nitely generated S-acts are congruence compact;

2) all cyclic S-acts are congruence compact;

3) S as a left S-act is congruence compact.

Proof. The implications 1)) 2) and 2)) 3) are trivial. 3)) 1). Let SA be a�nitely generated S-act and let � : SF ! SA be a surjective homomorphism where

SF is a �nitely generated free S-act. Now the implication follows from Lemma 2.2and Lemma 1.1.

If S is a right reversible monoid then the indecomposable components of acongruence compact act can be more precisely described. Namely, if S is a rightreversible monoid and SZ � SA a subact then Z = fa 2 Aj9s 2 S : sa 2 Zg is asubact of SA as well. For, if z 2 Z and s 2 S then there exists t 2 S such thattz 2 Z and, by right reversibility, there exist u; v 2 S such that us = vt . We haveu(sz) = (us)z = (vt)z = v(tz) 2 Z , i.e. sz 2 Z .

Now we have the following.

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Proposition 2.4. Every congruence compact act SA over a right reversible monoidis isomorphic to

`ni=1 Zi , where the Zi; i = 1; : : : ; n, are simple acts.

Proof. Let a 2 A be an arbitrary element. Then by right reversibility the setfSs1 : : : snag , where fs1; : : : ; sng is an arbitrary �nite subset of S , is a �lter baseof cyclic subacts of SA . Hence Za = \Ss1 : : : sna 6= �. If s 2 S and z 2 Za

then sz 2 sSs1 : : : sna � Ss1 : : : sna for every �nite subset fs1; : : : ; sng of S andtherefore sz 2 Za . This means that Za is a subact of SA . Let now z1; z2 2 Za .Then there exist u; v 2 S such that z1 = ua and z2 = va . Suppose z1 62 Sz2 .Then we have z1 62 Sz2 = Sva , a contradiction to the fact that z1 2 \Ss1 : : : sna ,where fs1; : : : ; sng is an arbitrary �nite subset of S . Hence Za is simple. Forevery s 2 S and for every �nite set fs1; : : : ; sng there exist u; v 2 S such thatus1 : : : sns = vs1 : : : sn and therefore Sus1 : : : snsa = Svs1 : : : sna � Ss1 : : : sna .Hence Zsa = \Ss1 : : : snsa � \Ss1 : : : sna = Za . By simplicity of Za we getthat Zsa = Za . If now sa = tb for some s; t 2 S , a; b 2 SA, then we haveZa = Zsa = Ztb = Zb . Therefore, if a subact SB 2 SA is connected then thesubacts Zb coincide for all b 2 B . Finally, by Lemma 2.1, SA has a �nite number ofconnected components.

Corollary 2.5. Every right reversible congruence compact monoid has a smallestleft ideal.

From Proposition 2.3 we see that congruence compact monoids (as acts) playa very important role in the theory of congruence compact acts. Therefore it wouldbe interesting to describe such monoids. In the following Proposition 2.9 a �rst stepis done in this direction. Before that we need two lemmas.

Lemma 2.6([3], Lemma 2.2). Let � =< tw; t > , where w; t 2 S . Then x�y holdsif and only if xwm = ywn for some m;n � 0, where xwi 2 St whenever 0 � i < m

and ywj 2 St whenever 0 � j < n .

We say that an element s 2 S is right quasiperiodic if there exist l 2 N and x 2 S

such that sl = xsl+1 . We also say that S is a right quasiperiodic monoid if all itselements are right quasiperiodic. It is obvious that s is right quasiperiodic if andonly if there exists l 2 N such that sl 2 \iSs

i .

Lemma 2.7. If fCigI is a �lter base of congruence classes of a congruence com-pact S-act SA then for every homomorphism � : SA ! SB we have �(\ICi) =\I�(Ci) .

Proof. We have to show \I�(Ci) � �(\ICi) only. Let x 2 \I�(Ci), i.e. x 2 �(Ci)for every i 2 I . Then ��1(x) \ Ci 6= � for every i 2 I , where ��1(x) = fz 2

SAj�(z) = xg . Therefore, f��1(x) \ CigI is a �lter base of congruence classesof SA . By congruence compactness there exists a z 2 \I(�

�1(x) \ Ci). Thenx = �(z) � �(\I (�

�1(x) \ Ci)) � �(\ICi).

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Corollary 2.8. If S is a (left) congruence compact monoid and fJi; i 2 Ig a�lter base of left ideals then (\IJi)s = \I(Jis) for every s 2 S .

A starting point in describing the structure of congruence compact monoidscould be the following.

Proposition 2.9. Every congruence compact monoid S (as a left S-act) is rightquasiperiodic.

Proof. Let S be congruence compact and let s 2 S . Let �k =< (sk; sk+1) > ,k = 1; 2; : : :. Obviously the congruence classes of �k containing sk form a �lterbase. Hence there exists x such that x�ks

k for all k . By Lemma 2.6 (takingt = sk and w = s) we have that for every k there exist m;n 2 N such thatxsm = sk+n and xsi 2 Ssk whenever 0 � i < m . Taking i = 0 we get thatx 2 Ssk for all k , i.e. x 2 \iSs

i . Denote \iSsi by Ts . Using Corollary 2.8

we get Ts = \1i=1(Ssi�1s) = (\1i=0Ss

i)s = Tss , and therefore Ts = Tssl for every

l 2 N . From xsm = sk+n and x 2 Ts we get now that sk+n 2 Ts and, in particular,sk+n 2 Ssk+n+1 .

If S is commutative then the last proposition can be generalized in the fol-lowing way:

Proposition 2.10. Let SA be a congruence compact act over a commutative mo-noid. Then for any two elements a 2 SA and s 2 S there exist t 2 S and l 2 N

such that sla = tsl+1a.

Proof. Let �i =< (sia; si+1a) > , i = 1; 2; : : : : Then the classes of �i containingsia form a �lter base. By congruence compactness of SA there exists a z 2 SA suchthat z�is

ia for all i 2 N . It is easy to show that x�iy if and only if x; y 2 Ssia andskx = sly for some k; l � i . Hence z 2 \iSs

ia , and skz = sla (because z�0a). Thensla = skz 2 \iSs

ia and, in particular, sla 2 Ssl+1a .

In [8, Proposition 6.3] it was proved that over a commutative monoid S; everycongruence compact act is 1-equationally compact. We proceed in this section with anexample showing that for general monoids, congruence compactness does not implyeven weak 1-equational compactness.

Example 2.11. Let A = fbigN[fcigN be a set of pairwise di�erent elements. LetS be the monoid of all mappings A! A . Then SA is congruence compact becausethere exist only trivial congruences on SA . For, let � =< (a; b) > , a 6= b , and letx 6= y be arbitrary elements of SA . Consider s 2 S such that sa = x and sz = y

for all z 2 Anfag . Then we have x = sa�sb = y . Let now ti 2 S , i = 1; 2; : : :, bede�ned as follows:

tia =

8<:ck+1; if a = ck;c0; if a = bi;a; if a = bk; k 6= i.

The S-act SA can be presented by the following labelled graph:

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@@R-

���

���

- - -

b0

b1

b2

c0 c1c2 :: :

t0

ti ti ti

t2

The system ft1x = x; t2x = x; : : : ; tnx = x; : : : g is �nitely solvable but notsolvable. This means that SA is not weakly 1-equationally compact.

Due to the fact that direct products of congruence compact acts are not necessarilycongruence compact (cf. Corollary 3.5) it is not possible to prove directly thatcongruence compact acts over a commutative monoid are equationally compact. Butwe are able to prove the following.

Proposition 2.12. Every congruence compact act over a commutative monoid isf-equationally compact.

Proof. Let SA be a congruence compact act over a commutative monoid S andlet � be a �nitely solvable system of equations on SA having n variables x1; : : : ; xn .We will prove that � is solvable in SA by using induction on n . If n = 1 thenthe proposition is proved in [8, Proposition 6.3]. Assume the proposition is true forn � 1 variables. Consider SB = �n

i=1SAi , SAi = SA. For every equation � 2 �of type sxi = a or of type sxi = txi de�ne B� = �n

j=1Bj , where Bi is the set ofall solutions of � and Bj = SA if j 6= i . By the proof of Proposition 6.3 of [8] itfollows that Bi is a congruence class of some congruence on SA and therefore B� isa congruence class of SB . With every equation � of type sxi = txj , i 6= j , relatea set B� = f(a1; : : : ; an)jsai = taj and ak is arbitrary if i 6= k 6= jg . Since B� isa subact of SB , B� is also a congruence class (of an appropriate Rees congruence).Further, for every �nite subset �0 � � denote B�0 = \�2�0B� . The set fB�0gobviously form a �lter base. Hence the set fprn(B�0)j�

0 - a �nite subset of �g ,where prn() is the projection on the nth component, form a �lter base (of congruenceclasses) on SAn = SA and has then a nonempty intersection \�0prn(B�0). Nowa 2 \�0prn(B�0) means that for every �nite subsystem �0 of � there exists a solutionwith xn = a . Hence, substituting xn by a everywhere in � we get a �nitely solvablesystem of equations in n � 1 variables. By hypothesis, this system has a solution,say (a1; : : : ; an�1). Then (a1; : : : ; an�1; an) is obviously a solution of �.

3. Congruence compact acts over monogenic monoids

Throughout this section, let S be a monogenic monoid, i.e. there exists s 2 S suchthat for every t 2 S we have t = sk for some k 2 N0 . An element a from SA issaid to have order n if there exists an element b 2 A such that a = snb . Denote thisby n = deg(b; a). Obviously, the set of orders of an element can be either �nite orin�nite. A subset Cn = fa1; : : : ; ang � SA is a cycle if sai = ai+1; i = 1; : : : ; n � 1;and san = a1 . For a subset C � SA de�ne

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deg(a; C) =

(minx2C

deg(a; x); if there exist n 2 N0; x 2 C such that x = sna;

1; otherwise.

Let SA be a connected S-act with a cycle Cn . De�ne a mapping �A : SA! Cnby putting

�A(a) = c 2 Cn; with skc = ska; where k = deg(a; Cn):

Obviously, �A is a correctly de�ned S-homomorphism and �A(c) = c for allc 2 Cn .

Lemma 3.1. Let C be a congruence class of some congruence � on a connectedS-act SA such that C \ Cn 6= � where Cn is a cycle. Then c��A(c) for all c 2 C .

Proof. Let a 2 C and let l 2 N be the minimal number such that sla 2 C (suchl exists because of C \ Cn 6= �). Then for c1; c2 2 Cn we have that c1�c2 if andonly if c1 = splc2 for some p 2 N . Let now m � deg(a; Cn) be the minimal naturalnumber such that m = m0l for some m0 2 N . We have a�sla� : : : �sma = �A(s

ma) =sm�A(a)��A(a).

Lemma 3.2 ([4, Paragraph 10, Theorem 4]). In a left S-act SA, (c; d) 2< H >

if and only if there exist an n 2 N, a sequence c = z0; z1; : : : ; zn = d of elementsof SA, pairs of elements (ai; bi) 2 H and elements si 2 S , i = 1; : : : ; n , such thatfsiai; sibig = fzi�1; zig , i = 1; : : : ; n.

Proposition 3.3. A connected S-act SA over a monogenic monoid is congruencecompact if and only if

1) SA contains a cycle Cn = fa1; : : : ; ang;

2) If B � SA is an in�nite subset then there exist b1; b2 2 B , b1 6= b2 , suchthat b1 = skb2 for some k 2 N .

Proof. Necessity. Assume SA is a congruence compact act over a monogenicmonoid S . If there exists a 2 A such that ska 6= sk+1a for all k 2 N , thenthe sets Ci = fskajk > ig , i 2 N , form a �lter base having empty intersection.Next we will show that every Ci can be considered as a congruence class of anappropriate congruence. For, let �i be a congruence generated by the set Ci � Ci

and let c�ix , where c 2 Ci . Then by Lemma 3.2 there exist x = z0; z1; : : : ; zn =c; kj 2 N; cj; c

0

j 2 Ci; j = 1; : : : ; n , such that fzj�1; zjg = skjfcj ; c0

jg . In particular,

for i = 1 we get fx; z1g = fz0; z1g = sk1fc1; c0

1g = sk1fsl1a; sl0

1ag for some l1; l0

1 > i .This means that x belongs to Ci . Assume now that B � SA is an in�nite subsetand there are no b1; b2 2 B , b1 6= b2 , such that b1 = skb2 for some k 2 N . Wewill show that every subset of B can be considered as a congruence class of somecongruence. For, let C � B be a subset and � a congruence generated by the setC �C . Let c�x where c 2 C . Then, using Lemma 3.2 again, we get that there existx = z0; z1; : : : ; zn = c , kj 2 N , cj ; c

0

j 2 C such that fzj�1; zjg = skjfcj; c0

jg . Take

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j = n . Then c = zn = skncn , from which we get kn = 0 and zn�1 2 C . Continuingin the same way we get x = z0 2 C .

Su�ciency. Let C = fCi; i 2 Ig be a �lter base of congruence classes of SA .It is clear that if some Ci is �nite then \ICi 6= �. Assume all Ci , i 2 I , are in�nite.By assumption, in every Ci; i 2 I , there are elements xi; yi 2 Ci and ki; i 2 N suchthat xi = skiyi . It is easy to see that then Ci \ Cn 6= � for every i 2 I . Denote byCk the subset of C consisting of all congruence classes Ci which do not contain theelement ak from the cycle Cn , k = 1; : : : ; n . Assume all Ck are nonempty. Take thenfrom each Ck one representative Dk; k = 1; : : : ; n . Let d 2 \n

1Dk . Then by Lemma3.1 we have that �A(d) 2 Cn belongs to all Dk; k = 1; : : : ; n , a contradiction. HenceCk = � for some k . This means that ak belongs to all Ci , i.e. the set \ICi is notempty.

From the last proposition we immediately get the following.

Corollary 3.4. All �nitely generated S-acts over a monogenic monoid S are con-gruence compact if and only if S is �nite.

In the category of all R-modules over a ring R , a direct product of congruencecompact R-modules is congruence compact ([10], Proposition 1). As we see from thefollowing corollary this is not the case in the category of S-acts.

Corollary 3.5. A direct product of two congruence compact S-acts is not neces-sarily congruence compact.

Proof. Let S be an in�nite monogenic monoid generated by s and let A =faiji 2 N; sa1 = a1 and sai+1 = ai for i > 1g . By Proposition 3.3, SA is congruencecompact. Then the set B = f(a2; ai)ji 2 Ng is an in�nite subset of SA � SA butthere are no pairs (a2; ai); (a2; aj); i 6= j , in B such that (a2; ai) = sk(a2; aj) forsome k . Hence SA� SA is not congruence compact, again by Proposition 3.3.

Let S be an in�nite monogenic monoid. By C11

denote the following S-act:fciji 2 Z; sci = ci+1g .

Lemma 3.6. ([9], Theorem 2). The unary algebra A = (A; s) is equationallycompact if and only if

1) for every a 2 A , if lim(n;n = order of a) = @0 then a is of in�nite order;

2) A contains either C11

or some Cn; n � 1.

Proposition 3.7. Every congruence compact S-act over a monogenic monoid isequationally compact.

Proof. By Lemma 3.6 we may assume that S is an in�nite monoid. Let SA becongruence compact and let a 2 A be such that lim(n;n = order of a) = @0 . From2) of Proposition 3.3 it follows that the set C1 = fc 2 Ajdeg(c; a) = 1g is �nite.Hence there exists c1 2 C1 such that lim(njn = order of c1) = @0 . Consider theset C2 = fc 2 Ajdeg(c; c1) = 1g . Analogously, there exists an element c2 2 C2

such that lim(njn = order of c2) = @0 . Repeating this we get an in�nite sequencec1; c2; c3; : : : ; with sci+1 = ci; i = 1; 2; 3; : : :, which means that a is of in�nite orderand SA is equationally compact by Lemma 3.6.

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4. Congruence compact acts over commutative groups

If G is a commutative group then G is obviously congruence compact (as a left G-act) if and only if G is congruence compact as a group. Such groups are describedin (cf. [7], Satz 2.4):

Lemma 4.1. A commutative group G is congruence compact if and only if G isa direct sum of a �nite number of groups of type Cpk or Cp1 .

Now, using Lemmas 1.1 and 2.1, the fact that every cyclic G-act over acommutative group G is a group again, and the fact that if a commutative group G

is isomorphic to a direct sum of a �nite number of groups Cpk or Cp1 , then G hasthe minimal condition for congruences, we get the following.

Proposition 4.2. For a G-act GA over a commutative group, the following con-ditions are equivalent

1: GA is congruence compact;

2: GA has the minimal condition for congruences;

3: GA is isomorphic to a direct sum of a �nite number of groups Cpk or Cp1 .

Equationally compact G-acts over arbitrary groups are described in [2]. For agroup, the notions "equationally compact G-act" and "weakly 1-equationally compactG-act" coincide and therefore we immediately get from [8, Proposition 6.3] (or fromProposition 2.12) the following.

Corollary 4.3. All congruence compact G-acts over a commutative group G areequationally compact.

The author thanks the Deutscher Akademischer Austauschdienst for �nancialsupport and the Carl von Ossietzky University of Oldenburg Mathematics Depart-ment for excellent working conditions.

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[9] Wenzel, G.H., Subdirect irreducibility and equational compactness in unaryalgebras < A; f > , Archiv der Math.(Basel) 21 (1970), p.256{264.

[10] Zelinsky, D., Linearly compact modules and rings, Amer.J.Math. 75(1953),p. 79{90.

Tallinna Pedagoogika�ulikoolNarva mnt. 25EE0100 TallinnEstoniae{mail: pnormak@tpedi.estnet.ee

Received January 31, 1995and in �nal form December 31, 1995

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