Partial algebras and implications of (weak) matrix properties

TL;DR

This paper explores the relationships between various matrix properties in categories, showing that implications between these properties can be understood through partial terms and their weak versions, simplifying the analysis of their logical structure.

Contribution

It demonstrates that implications between matrix properties reduce to constructing partial terms and establishes their equivalence with implications among weak versions of these properties.

Findings

Implication checking reduces to partial term construction.

Implications among properties are equivalent to those among their weak versions.

Provides a new perspective on the logical structure of matrix properties.

Abstract

Matrix properties are a type of property of categories which includes the ones of being Mal'tsev, arithmetical, majority, unital, strongly unital and subtractive. Recently, an algorithm has been developed to determine implications $M\Rightarrow N$ between them. We show here that this algorithm reduces to construct a partial term corresponding to $N$ from a partial term corresponding to $M$. Moreover, we prove that this is further equivalent to the corresponding implication between the weak versions of these properties, i.e., the one where only strong monomorphisms are considered instead of all monomorphisms.