Unarily Extended Semilattices

Abstract

In this paper we give equational and structural characterizations of all algebras representing the sequence (0, 3, 1, 1, . . . ). In the first part of the paper we prove that an algebra without constants has exactly three distinct essentially unary term operations and exactly one essentially n-ary term operation for every n > 1 if and only if it is clone equivalent to a generic of one of three varieties of algebras defined in Section 2. The second part of the paper is devoted to the representations of algebras from these three varieties by means of semilattice ordered systems of algebras.

Subdirect Product Representations of Some Unary Extensions of Semilattices

“…The structure of algebras from the varieties V£ for k E {1,2,3} is clear, since / yields a retraction in algebras from Vjf, / yields a permutation of order 2 in algebras from V£ and / yields a mapping of order at most 2, i.e., f^(x) ~ /I 2 '(a;) « χ 2 , in algebras from V3, cf. [4]. However, for considerations in Sections 3 and 4 we need more detailed description of algebras from V£, which we include below.…”

“…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.…”

“…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.…”