A Pierce Representation Theorem for varieties with BFC

TL;DR

This paper extends the Pierce representation theorem to a broader class of algebraic categories with Definable Factor Congruences, using topos theory, and characterizes coextensive categories with stable centers by complements.

Contribution

It generalizes the Pierce representation theorem to algebraic categories with Definable Factor Congruences and provides a new characterization of coextensive categories.

Findings

Generalization of Pierce representation theorem to new algebraic categories

Characterization of coextensive categories with stable centers

Use of topos theory in algebraic structure analysis

Abstract

We generalize the Pierce representation theorem for (commutative) rings with unit to other algebraic categories with Definable Factor Congruences by using tools from topos theory. Of independent interest, we prove that an algebraic category with right existential definable factor congruences is coextensive if and only if has center stable by complements.