On the internal characterization of injective algebras

TL;DR

This paper characterizes injective universal algebras through an internal property called completeness, showing their equivalence and providing examples like Boolean algebras and divisible Abelian groups.

Contribution

It introduces a new internal property called completeness that characterizes injective algebras within their equational classes.

Findings

Injective algebras are characterized by the property of completeness.

Complete algebras are injective within their classes.

Boolean algebras and divisible Abelian groups are examples of complete algebras.

Abstract

It is shown that universal algebras that are injective in their equational classes are characterized by internal property that can be called completeness. We define universal algebra $A$ as complete (closed to simple extensions) if for each its subalgebra $A'$ and each set of extension conditions for this subalgebra there is $a \in A$ that satisfies these conditions. We define a set of extension conditions for $A'$ to some extension $A''$ as the difference between factorization kernels of free algebras for $A''$ and $A'$. It's proved that each injective universal algebra is complete and each complete universal algebra belonging to the class of algebras with CEP is injective. It's checked directly that complete (in the sense of ordering) boolean algebras and divisible Abelian groups are complete in the sense defined here.