Subdirect Product Representations of Some Unary Extensions of Semilattices

Abstract

Abstract. An algebra 21 represents the sequence so = (0,3,1,1,...) if there are no constants in 21, there are exactly 3 distinct essentially unary polynomials in 2t and exactly 1 essentially n-ary polynomial in 21 for every η > 1. It was proved in [4] that an algebra 21 represents the sequence so if and only if it is clone equivalent to a generic of one of three varieties Vi, V2, V3, see Section 1 of [4]. Moreover, some representations of algebras from these varieties by means of semilattice ordered systems of algebras were given in [4], In this paper we give another, by subdirect products, representation of algebras from Vi, V2, V3. Moreover, we describe all subdirectly irreducible algebras from these varieties and we show that if an algebra 21 represents the sequence so, then it must be of cardinality at least 4. PreliminariesBy an algebra we mean a pair 21 = (A; F α ), where A is a nonempty set called the carrier of 21 and F 21 is a set of finitary operations in 21 called the set of fundamental operations of 21. By the clone of 21 we mean as usually the smallest set containing all projections and closed under superpositions with fundamental operations of 21. We denote it by CJ (21). The operations from CI (21) are called polynomials, see [2], Two algebras 21 = (A\ F*) and 93 = (B; F®) we call clone equivalent if A = Β and Ci (21) = Cl (05). A polynomial / a (xi, ..., x n ) depends on the variable x k , k G {1, ..., η}, if there exist αϊ, ..., a n , b G A such that / a (ai, ..., a n ) φ / α (αι, ..., a k -1, b, a k+ 1, ..., α"). A polynomial 1, ..., x n ) is essentially n-ary, if it depends on each of its variables. We denote by p n (21) the cardinality of the set of all essentially n-ary polynomials of 21, if η > 0. Next, we denote by po(2l) the cardinality of the set of all constant unary polynomials of 21, see [3].2000 Mathematics Subject Classification: Primary 06E05; Secondary 20N02, 08A05, 08B26.