Extension Monads: Some Structure Theorems

TL;DR

This paper explores the structure of extension monads in algebraic categories, demonstrating their behavior with respect to operations like direct products and retractions, and establishing key structural theorems.

Contribution

It provides new structure theorems for extension monads, showing their commutation with direct products and the triviality of retractions in algebraic contexts.

Findings

Extension monads commute with direct product operations.

Only trivial retractions exist from an extension monad.

Structural theorems clarify the behavior of extension monads in algebraic categories.

Abstract

We investigate the behavior of extension monads, introduced in the 1990s by the second author, in terms of structure results for infinitely many finitary operations and common constructions in varieties or categories of algebras. Specifically, we see that for finite collections of algebras of the same signature there the extension monad operation commutes with the direct product operation, and that the only retractions from an enlargement or extension monad are trivial.