# Epimorphism surjectivity in varieties of Heyting algebras

## TL;DR

This paper explores the conditions under which epimorphisms are surjective in varieties of Heyting algebras, providing new examples, counterexamples, and characterizations that clarify the relationship between algebraic properties and logical definability.

## Contribution

It identifies a continuum of non-surjective epimorphisms in certain Heyting algebra varieties and characterizes subvarieties with surjective epimorphisms, advancing understanding of algebraic and logical properties.

## Key findings

- A continuum of non-surjective epimorphisms in the variety generated by the Rieger-Nishimura lattice.
- Varieties of finite width have non-surjective epimorphisms for n ≥ 2.
- Characterization of subvarieties in the Kuznetsov-Gerciu variety with surjective epimorphisms.

## Abstract

It was shown recently that epimorphisms need not be surjective in a variety K of Heyting algebras, but only one counter-example was exhibited in the literature until now. Here, a continuum of such examples is identified, viz. the variety generated by the Rieger-Nishimura lattice, and all of its (locally finite) subvarieties that contain the original counter-example K. It is known that, whenever a variety of Heyting algebras has finite depth, then it has surjective epimorphisms. In contrast, we show that for every integer n greater or equal than 2, the variety of all Heyting algebras of width at most n has a non-surjective epimorphism. Within the so-called Kuznetsov-Gerciu variety (i.e., the variety generated by finite linear sums of one-generated Heyting algebras), we describe exactly the subvarieties that have surjective epimorphisms. This yields new positive examples, and an alternative proof of epimorphism surjectivity for all varieties of Goedel algebras. The results settle natural questions about Beth-style definability for a range of intermediate logics.