On the structure of balanced residuated partially ordered monoids

TL;DR

This paper explores the structure of balanced residuated partially ordered monoids, providing a decomposition method and a construction approach that enhances understanding of their varieties and related algebraic systems.

Contribution

It generalizes Plonka sum constructions to balanced residuated posets and introduces a new method for constructing residuated posets from semilattice directed systems.

Findings

Balanced residuated posets can be decomposed into sums indexed by positive idempotent elements.

A new construction method for residuated posets from semilattice directed systems.

Structural descriptions of certain varieties of residuated lattices and relation algebras.

Abstract

A residuated poset is a structure $\langle A,\le,\cdot,\backslash,/,1 \rangle$ where $\langle A,\le \rangle$ is a poset and $\langle A,\cdot,1 \rangle$ is a monoid such that the residuation law $x\cdot y\le z\iff x\le z/y\iff y\le x\backslash z$ holds. A residuated poset is balanced if it satisfies the identity $x\backslash x \approx x/x$. By generalizing the well-known construction of Plonka sums, we show that a specific class of balanced residuated posets can be decomposed into such a sum indexed by the set of positive idempotent elements. Conversely, given a semilattice directed system of residuated posets equipped with two families of maps (instead of one, as in the usual case), we construct a residuated poset based on the disjoint union of their domains. We apply this approach to provide a structural description of some varieties of residuated lattices and relation algebras.