A categorical description of simple Beth companions

TL;DR

This paper characterizes simple Beth companions of quasivarieties through categorical properties, showing their uniqueness and connection to mono-reflective subcategories.

Contribution

It provides a categorical framework for understanding simple pp expansions and characterizes simple Beth companions uniquely via monomorphisms.

Findings

Simple pp expansions coincide with certain quasivarieties with well-defined forgetful functors.

A simple Beth companion, if it exists, is unique up to term equivalence.

Categorical conditions characterize simple Beth companions as mono-reflective subcategories.

Abstract

A pp expansion of a quasivariety $\mathsf{K}$ is said to be simple when it is of the form $\mathsf{K}[\mathscr{L}_\mathcal{F}]$. For instance, when $\mathsf{K}$ has the amalgamation property, all its pp expansions are simple. It is shown that the simple pp expansions of a quasivariety $\mathsf{K}$ coincide with the quasivarieties $\mathsf{M}$ for which the forgetful functor $U \colon \mathsf{M} \to \mathsf{K}$ is well defined and induces an isomorphism from $\mathsf{M}$ to a mono-reflective subcategory of $\mathsf{K}$. As a consequence, if a quasivariety $\mathsf{K}$ possesses a simple Beth companion $\mathsf{M}$, then $\mathsf{M}$ is the unique (up to term equivalence) quasivariety whose monomorphisms are regular that, moreover, satisfy the categorical description of simple pp expansions of $\mathsf{K}$ given above.