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Subdirect Irreducibility of Algebras and Acts with an Additional Unary Operation

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Subdirect Irreducibility of Algebras and Acts with an Additional Unary Operation 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'.
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