On Regular Congruences of Ordered Semigroups

An ordered semigroup is a structure S = 〈S, ·,≤〉 with a binary operation · that is associative and a partial ordering ≤ that is compatible with the binary operation. For a given congruence relation θ of the semigroup S = 〈S, ·〉 the quotient structure S/θ = 〈S/θ, , 〉 is not in general an ordered semigroup. In this paper we study quotients of ordered semigroups. We first define a special type of congruences, called regular congruences, that will preserve ordering on the quotient structures. We then show that the set of all regular congruences with the ordering ≤ is an algebraic lattice. Afterwards, we discuss the link between finitely generated regular congruences and subdirectly irreducible ordered semigroups. At the end we will discuss generalization of these concepts to an arbitrary ordered algebra. Definition 1. A partially ordered semigroup (in the remainder of the text posemigroup) is an ordered triple S = 〈S, ·,≤〉 such that a) 〈S, ·, 〉 is a semigroup, b) 〈S, ≤〉 is a partially ordered set (briefly poset), c) (x ≤ y & u ≤ v)→ x · u ≤ y · v for all x, y, u, v ∈ S. Definition 2. A congruence relation of the semigroup S = 〈S, ·〉 is an equivalence relation θ ⊆ S that satisfies the following compatibility property: (xθy & uθv)→ x · u θ y · v for all x, y, u, v ∈ S. (CP) Given a congruence relation θ of the semigroup S = 〈S, ·〉 (or an algebra in general) we define the quotient semigroup (a.k.a the homomorphic image) of S by θ to be the semigroup S/θ = 〈S/θ, 〉 where x/θ y/θ = (x · y)/θ for all x, y ∈ S. The set of all congruences of a semigroup S = 〈S, ·〉 is denoted by Con(S). How does one extend the concept of quotients to po-semigroups? In general, for structures that have only operations, i.e., algebras, quotients are defined using congruences. Structures that have both operations and relations are studied in model theory and their quotients are defined as The algebraic quotient + Relations on the algebraic quotient. In case of the po-semigroups we have the following definition of the quotient.