Relational Semigroups and Object-Free Categories

TL;DR

This paper explores the connections between axioms for partial semigroups, monoids, and small object-free categories, examining their structures, adjunctions, and providing examples to distinguish different algebraic frameworks.

Contribution

It establishes relationships between partial semigroup axioms and object-free categories, including the adjunction of zero elements and illustrative examples of algebraic distinctions.

Findings

Axioms for partial semigroups relate to small object-free categories.

Adjunction of zero elements affects algebraic structures.

Examples demonstrate separation of algebraic classes.

Abstract

This note relates axioms for partial semigroups and monoids with those for small object-free categories, either with multiple monoidal units or with source and target maps. We discuss the adjunction of a zero element to both kinds of category and provide examples that separate the algebras considered.