Epic substructures and primitive positive functions

TL;DR

This paper characterizes epic substructures in classes of structures closed under ultraproducts and applies the results to quasivarieties with near-unanimity terms, establishing conditions for surjective epimorphisms and their decidability.

Contribution

It provides a characterization of epic substructures via primitive positive formulas and applies this to analyze surjective epimorphisms in quasivarieties with near-unanimity terms.

Findings

Epic substructures are characterized by primitive positive formulas.

Surjective epimorphisms in certain quasivarieties are characterized by specific algebraic conditions.

Decidability of surjective epimorphisms is established for finite sets of finite algebras with near-unanimity terms.

Abstract

For $\mathbf{A}\leq\mathbf{B}$ first order structures in a class $\mathcal{K}$, say that $\mathbf{A}$ is an epic substructure of $\mathbf{B}$ in $\mathcal{K}$ if for every $\mathbf{C}\in\mathcal{K}$ and all homomorphisms $g,g^{\prime}:\mathbf{B}\rightarrow\mathbf{C}$, if $g$ and $g'$ agree on $A$, then $g=g'$. We prove that $\mathbf{A}$ is an epic substructure of $\mathbf{B}$ in a class $\mathcal{K}$ closed under ultraproducts if and only if $A$ generates $\mathbf{B}$ via operations definable in $\mathcal{K}$ with primitive positive formulas. Applying this result we show that a quasivariety of algebras $\mathcal{Q}$ with an $n$-ary near-unanimity term has surjective epimorphisms if and only if $\mathbb{SP}_{n}\mathbb{P}_{u}(\mathcal{Q}_{RSI})$ has surjective epimorphisms. It follows that if $\mathcal{F}$ is a finite set of finite algebras with a common near-unanimity term, then it is decidable whether the (quasi)variety generated by $\mathcal{F}$ has surjective epimorphisms.