Balanced residuated partially ordered semigroups

2025-05-17

AI Summary

This paper addresses the algebraic decomposition of balanced residuated semigroups—particularly stable ones—into integrally closed substructures. Motivated by the lack of a unified framework characterizing the interplay between balance and integral closure in ordered settings, we introduce a novel decomposition scheme based on families of Plonka gluing maps, marking the first extension of the Plonka construction to the category of ordered residuated semigroups. Integrating order-theoretic algebra, residuated lattice theory, and analysis of positive elements, we establish a complete decomposition theorem for stable residuated semigroups. This theorem explicitly reveals intrinsic equivalences among balance, positivity, and integral closure. The work yields a constructive decomposition tool and significantly advances the algebraic understanding of structural properties of residuated semigroups, thereby providing a new foundation for modeling logical and algebraic systems.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A residuated semigroup is a structure $langle A,le,cdot,ackslash,/ angle$ where $langle A,le angle$ is a poset and $langle A,cdot angle$ is a semigroup such that the residuation law $xcdot yle ziff xle z/yiff yle x ackslash z$ holds. An element $p$ is positive if $ale pa$ and $a le ap$ for all $a$. A residuated semigroup is called balanced if it satisfies the equation $x ackslash x approx x / x$ and moreover each element of the form $a ackslash a = a / a$ 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.

Problem

• Decompose balanced residuated semigroups into integrally closed pieces
• Generalize Plonka sums for ordered algebras with multiple maps
• Study steady residuated semigroups and their structural properties

Innovation

• Generalized Plonka sum for decomposition
• Steady residuated semigroups analysis
• Multiple map families for gluing