View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of the Academy's Library Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 6 (2005), No 2, pp. 217-224 DOI: 10.18514/MMN.2005.114 Subdirect irreducibility of algebras and acts with an additional unary operation N.V. Loi and R. Wiegandt Miskolc Mathematical Notes HU ISSN 1787-2413 Vol. 6 (2005), No. 2, pp. 217–224 electronic version SUBDIRECT IRREDUCIBILITY OF ALGEBRAS AND ACTS WITH AN ADDITIONAL UNARY OPERATION N. V. LOI AND R. WIEGANDT [Received: December 12, 2004] Abstract. As a generalization of algebras with involution [1], we consider algebras (A; F [ f'g) with an additional unary operation ' of finite degree k. A subdirectly irreducible algebra (A; F [ f'g) is a subdirect product of subdirectly irreducible algebras (A='(Θ); F );:::; (A='d(Θ); F ) where Θ is a certain congruence, d is a divisor of k and 'd is the identical mapping. For a congruence modular algebra (A; F ) also the unique smallest congruence of (A; F [ f'g) described. Examples are S -acts with an additional unary operation. In this case more information is given on the subdirectly irreducible components. Mathematics Subject Classification: 08A05, 08A60 Keywords: subdirectly irreducible algebra, congruence, unary operation, S -act 1. Preliminaries The purpose of this note is to generalize the notion of involution of universal al- gebras and to study the structure of subdirectly irreducible universal algebras with generalized involution, that is, with an additional unary operation of finite degree. Let ' be a variety of universal algebras (A; F [ f'g) with finitary operations f 2 F and a unary operation ' : a ! '(a) subject to the identities (i) Eλ = Eλ( fλ1 ;:::; fλnλ ) j fλ1 ;:::; fλnλ 2 F ), λ 2 Λ, involving only operations from F , k (ii) E' : ' (a) = a, ≥ ' where k 2 is a fixed integer, the degree of , (iii) E f : ' f (a1;:::; an) = f ' aπ f (1);:::;'(aπ f (n)) for all f 2 F , where π f is a permutation of n elements (n the arity of the operation f ) assigned to each operation f 2 F . Each permutation π f can be written into a product of disjoint cyclic permutations c(π f )1;:::; c(π f )r admitting also cyclic permutations of length 1. These identities may induce some further ones. The identity f = 'k( f ) is a conse- quence of (ii), and this is not trivial if and only if the degree k of ' is not a multiple Research supported by OTKA, Grant No. T043034. c 2005 Miskolc University Press 218 N. V. LOI AND R. WIEGANDT k of the lengths c(π f )1;:::; c(π f )r. The identity f = ' ( f ) may be viewed as a gener- alized commutativity; in the case of a binary operation f (x1; x2), cyclic permutation c(π f ) = (12) = π f and odd degree k = 2` + 1 we have k 2`+1 f (x1; x2) = ' ( f (x1; x2)) = ' ( f (x1; x2)) = '( f (x1; x2)) = f (x2; x1); that is, the commutativity of the operation f . Moreover, if f :! a and g : a ! g(a) are 0-ary and unary operations, then by (iii) it follows that '( f ):! a and '(g(a)) = g('(a)): In particular, for the zero element 0 and the unity element 1 (if they exist) we have '(0) = 0 and '(1) = 1; and also '(−a) = −'(a) for the additive inverse of an element a. We are going to scrutinize the interrelations between the variety ' and the variety of algebras (A; F ) subject to the identities (i). Let us observe that for any algebra (A; F [ f'g) and congruence Θ 2 Con(A; F ) of the algebra (A; F ) 2 also '(Θ) = ('(a);'(b)) j (a; b) 2 Θ is a congruence of (A; F ), as seen readily from (iii). Proposition 1. (a) Let (A; F [ f'g) 2 ' and Θ 2 Con(A; F ). Then Θ = '(Θ) if and only if Θ 2 Con(A; F [ f'g). (b) The lattice Con(A; F [ f'g) is a sublattice of the lattice Con(A; F ). (c) If Θ 2 Con(A; F ), then '(Θ) _···_ 'k(Θ) and '(Θ) ^ · · · ^ 'k(Θ) belong to Con(A; F [ f'g). Proof. The statements correspond to those of [1, Proposition 3.1] and their proofs are straightforward. As a motivation for the subsequent investigations we give an example. Example. Let G be an abelian group, and consider the direct sum A = G ⊕ G. On A we define a unary operation ' by '(a; b) = (b; a + b)(8a; b 2 G): Since '((a; b) + (c; d)) = (b + d; a + b + c + d) = '(a; b) + '(c; d); the operation fulfils the requirement of condition (iii). One readily sees that if A is an elementary 2-group, then '3(a; b) = (a + 2b; 2a + 3b) = (a; b); SUBDIRECT IRREDUCIBILITY OF ALGEBRAS 219 so the degree of ' is 3. Clearly, (0; G) defines a congruence of (A; +), '(0; G) = f(a; a) j a 2 Gg gives the diagonal congruence of A and '2(0; G) is the congruence determined by (G; 0). If jGj = 2, then all the three factors of (A; +) by these congruences are sub- directly irreducible, and (A; +) is a subdirect product of these factors. Nevertheless, (A; f+;'g) itself is subdirectly irreducible. For elementary 3−; 5−; 7−; 11− groups, the degree of ' is 8, 20, 16, 10, respec- tively. If G is a cyclic group of 4 elements or a direct sum of such groups, then the degree of ' is 12. 2. Subdirect decompositions As seen in [1], a subdirectly irreducible algebra (A; F [f'g) with involution ' (that is the degree of ' is k = 2) is either subdirectly irreducible as an algebra (A; F ) 2 or a certain subdirect product of two subdirectly irreducible algebras of . We shall generalize this result, and describe the subdirectly irreducible algebra (A; F [ f'g) 2 ' as a subdirect product of subdirectly irreducible algebras of and see that the number d of the subdirect components is a divisor of the degree k of '. The smallest congruence of an act A will be denoted by !A or briefly by !. We shall call the unique atom of the lattice of congruences of a subdirectly irreducible algebra the heart of that algebra. This terminology corresponds to that of ring theory: the unique atom in the lattice of ideals of a subdirectly irreducible ring used to be called the heart of that ring. Theorem 2. If (A; F [f'g) is a subdirectly irreducible algebra with heart χ', then there exists a congruence Θ 2 Con(A; F ) such that χ' Θ and (A; F ) is isomorphic to a subdirect product of subdirectly irreducible algebras (A='(Θ); F ); (A='2(Θ); F );:::; (A='d(Θ); F ); (1) where d is the least positive integer such that 'd(Θ) = Θ. The heart of (A='i(Θ); F ) is χ = (χ _ 'i(Θ))='i(Θ) for each i = 1;:::; d. The isomorphism between (A; F ) i ' N d i and the subdirect product i=1(A=' (Θ); F ) is given by the correspondence a 7−! ([a]'(Θ);:::; [a]'d(Θ))(8a 2 A); i where [a]'i(Θ) stands for the ' (Θ)-coset represented by a 2 A. Proof. If (A; F ) is subdirectly irreducible, then d = 1 and Θ = ! = '(!) = '(Θ) does the job. Suppose, next, that (A; F ) is not subdirectly irreducible. Since χ' 2 Con(A; F ), there exists a congruence Θ 2 Con(A; F ) such that ! , Θ , '(Θ), χ' * Θ and by an appropriate choice of Θ it can be achieved (Zorn’s Lemma) that (A=Θ; F ) is subdirectly irreducible and in this case the heart of (A=Θ; F ) is obviously χ1 = d (χ' _ Θ)=Θ. Let d be the smallest integer such that ' (Θ) = Θ. Such an integer d 220 N. V. LOI AND R. WIEGANDT does exist, and d must divide k, the degree of '. Moreover, by (ii) also (A='i(Θ); F ) is subdirectly irreducible with heart χi for every i = 1;:::; d. Since χ' * Θ and '(Θ) ^ '2(Θ) ^ · · · ^ 'd(Θ) is in Con(A; F [ f'g), we conclude that '(Θ) ^ '2(Θ) ^ · · · ^ 'd(Θ) = !: Thus, (A; F ) has the desired subdirect product representation. The last assertion is obvious. In the next two theorems we shall use the assumption that the congruence lattice of the algebra (A; F ) is modular. This is always so for groups, rings, and in many other cases, but in general the congruence lattice of an algebra, in particular of an S -act, need not be modular. Remark. In [1, Theorem 3.2] it has to be assumed that the congruence lattice of (A; F ) is modular (this was used on line 8 of its proof). This inaccuracy, however, does not affect the further results of [1]. Theorem 3. Let (A; F [ f'g) be an algebra as in Theorem 2. If (A; F ) is con- gruence modular and χ' ^ Θ = !, then χ' is an atom in Con(A; F ) and there is a one-to-one correspondence between the χ'-cosets of (A; F ) and the χi-cosets of (A='i(Θ); F ) given by the correspondence [a]χ' 7−! [[a]'i(Θ)]χi ; which can be expressed also in terms of relations as aχ'b 7−! [a]'i(Θ)χi[b]'i(Θ): Proof. If d = 1, then there is nothing to prove. Suppose that d > 1, and let κ 2 Con(A; F ) be a congruence such that κ < χ'.