Balanced residuated partially ordered semigroups
Stefano Bonzio, José Gil-Férez, Peter Jipsen, Adam Přenosil, Melissa Sugimoto
TL;DR
The paper addresses the structural analysis of balanced residuated semigroups by decomposing them into integrally closed fibers indexed by central positive idempotents, using a generalized Płonka sum based on semilattice-directed systems of metamorphisms. It develops the theory of metamorphisms, partition systems, and partition pairs to glue fibers together, yielding a reversible correspondence between steady residuated semigroups and sums of integrally closed residuated monoids. In the idempotent case, the fibers become Brouwerian algebras and the framework recovers commutativity, with implications for lattice-ordered structures. The work is supported by instructive examples including residuated posets and relation algebras, and opens avenues for extending these decompositions to broader varieties and iterated constructions.
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.