Completeness (closedness to simple extensions) as a general concept in universal algebra.
Pavlo Dzikovsky · arXiv (Cornell University) · 2021
It is shown that categorical injectivity of complete boolean algebras and complete (divisible) Abelian groups is based on the common algebraic property that can be defined as a completeness of universal algebra. We define universal algebra $A$ as complete (closed to simple extensions) if for each its subalgebra and arbitrary set of extension conditions for this subalgebra there is $a \in A$ that satisfies these conditions. We define the set of extension conditions as the difference between factorization kernels of free algebras for the extended and source (sub)algebra. It's proved that universal algebra is complete if and only if it is an injective object in its equational class.