Balanced residuated partially ordered semigroups

TL;DR

This paper introduces balanced residuated semigroups, a class of algebraic structures, and demonstrates their decomposition into integrally closed components using a generalized Plonka sum construction.

Contribution

It defines balanced residuated semigroups and extends the Plonka sum method to decompose them into integrally closed parts.

Findings

Balanced residuated semigroups can be decomposed into integrally closed components.

The generalized Plonka sum involves multiple families of maps for gluing algebras.

The decomposition provides new insights into the structure of these 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.