Some structural and residual properties of 2-semilattices
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Algebra Universalis
Some structural and residual properties of 2-semilattices Ian Payne Abstract. To each 2-semilattice, one can associate a digraph and a partial order. We analyze these two structures working toward two main goals: One goal is to give a structural dichotomy on minimal congruences of 2semilattices. From this, we are able to deduce information about the tamecongruence-theoretic types that occur in 2-semilattices. In particular, we show that the type of a finite simple 2-semilattice is always either 3 or 5 and can be deduced immediately from its associated digraph. The other goal is to introduce and explore a property that some 2-semilattices have which we have named the “component-semilattice property”. We show that this property must hold in every algebra in a variety of 2-semilattices that is both locally finite and residually small. Hence, a finite 2-semilattice that lacks this property generates a residually large variety. Mathematics Subject Classification. 08A05, 08B26. Keywords. 2-Semilattices, Digraphs, Subdirectly irreducible, Residually large varieties, Meet-semidistributivity.
1. Introduction The earliest mention of 2-semilattices was by Quackenbush in [18]. There, a 2-semilattice was defined to be an algebra with one basic operation which is binary and satisfies all at-most-two variable identities that hold in the variety of semilattices. Quackenbush did not discuss 2-semilattices beyond stating that a 2-semilattice operation generates a minimal clone. While algebraic structures less general than 2-semilattices have been studied since as early as the 1970s, the first author to focus on 2-semilattices in general was Bulatov [4]. Bulatov’s definition of a 2-semilattice was an algebra A having only one basic operation ∗ Presented by A. A. Bulatov. Thanks to the Department of Mathematics and Statistics at McMaster University for their support in preparing this paper. 0123456789().: V,-vol
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I. Payne
Algebra Univers.
which is binary, commutative, idempotent, and satisfies x∗(x∗y) ≈ x∗y. Seeing that the definitions of Bulatov and Quackenbush are equivalent comes down to showing that the two-generated free 2-semilattice is a semilattice. We formally state and prove the equivalence of the two definitions in Proposition 3.2 since there does not appear to be a proof anywhere in the literature. One of the two main goals of this paper is to collect some general facts about 2-semilattices. As Bulatov [4] and Mar´ oti [14] did, we will explore 2semilattices largely through the lens of a digraph that can naturally be associated to each 2-semilattice. The digraph associated to a 2-semilattice A has domain A and an arrow from a to b when a ∗ b = b. The transitive closure of this relation is a quasiorder and gives rise to a partial order on A/ΔA , where ΔA is the equivalence relation whose classes are the strongly connected components of the digraph. In Section 3, we collect some basic properties of these digraphs and partial orders. In Section 4, we give a structural dichotomy in terms of the
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