Extension Monads: Some Structure Theorems
Danielle Bowerman, Matt Insall

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.
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Taxonomy
TopicsAdvanced Algebra and Logic
