Strict refinement property of connected loop-free categories

TL;DR

This paper extends Hashimoto's Theorem to connected loop-free categories, demonstrating that product isomorphisms are uniquely determined by factor isomorphisms, leading to minimal decompositions of such structures.

Contribution

It generalizes the strict refinement property from partial orders to connected loop-free categories, broadening the theorem's applicability.

Findings

Hashimoto's theorem extended to connected loop-free categories.

Product isomorphisms are uniquely determined by factor isomorphisms.

Provides minimal decomposition for connected loop-free categories.

Abstract

In this paper we study the strict refinement property for connected partial ordersalso known as Hashimoto's Theorem. This property implies that any isomorphismbetween products of irreducible structures is determined is uniquely determinedas a product of isomorphisms between the factors. This refinement implies asort of smallest possible decomposition for such structures. After a brief recallof the necessary notion we prove that Hashimoto's theorem can be extendedto connected loop-free categories, i.e. categories with no non-trivial morphismsendomorphisms. A special case of such categories is the category of connectedcomponents, for concurrent programs without loops.