On Regular Congruences of Ordered Semigroups

Abstract. 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 struc-tures. We then show that the set of all regular congruences with the ordering ≤ is an algebraic lattice. Afterwards, we discuss the link be-tween 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 po-semigroup) is an ordered triple S = 〈S, ·,≤ 〉 such that a) 〈S, ·, 〉 is a semigroup