Relative ideals in homological categories, with an application to MV-algebras

TL;DR

This paper introduces the concept of relative ideals in homological categories, demonstrating their relevance through properties of MV-algebras and related structures, and applies these ideas to categories like hoops and MV-algebras.

Contribution

It defines relative ideals in homological categories and shows how objects can be viewed as such ideals, with applications to MV-algebras and related algebraic structures.

Findings

Category of hoops is semi-abelian

Category of MV-algebras is protomodular

Application to the forgetful functor from MV-algebras to Wajsberg hoops

Abstract

Let $A$ be a homological category and $U\colon B\to A$ be a faithful conservative right adjoint. We introduce the notion of relative ideal with respect to $U$, and we show that, under suitable conditions, any object of $A$ can be seen as a relative ideal of some object in $B$. We then develop a case study. We first prove that the category of hoops is semi-abelian and that the category of MV-algebras is protomodular, then we apply our results to the forgetful functor from the category of MV-algebras to the category of Wajsberg hoops.