Balanced residuated partially ordered semigroups

Abstract

A residuated semigroup is a structure $\langle A,\le,\cdot,\backslash,/ \rangle$ where $\langle A,\le \rangle$ is a poset and $\langle A,\cdot \rangle$ is a semigroup such that the residuation law $x\cdot y\le z\iff x\le z/y\iff y\le x \backslash z$ holds. An element $p$ is positive if $a\le pa$ and $a \le ap$ for all $a$. A residuated semigroup is called balanced if it satisfies the equation $x \backslash x \approx x / x$ and moreover each element of the form $a \backslash a = a / a$ is positive, and it is called integrally closed if it satisfies the same equation and moreover each element of this form is a global identity. We show how a wide class of balanced residuated semigroups (so-called steady residuated semigroups) can be decomposed into integrally closed pieces, using a generalization of the classical Plonka sum construction. This generalization involves gluing a disjoint family of ordered algebras together using multiple families of maps, rather than a single family as in ordinary Plonka sums.