Epimorphisms in varieties of residuated structures

TL;DR

This paper investigates when epimorphisms are surjective in various residuated structures, establishing conditions for the Beth definability property in substructural logics and highlighting differences between finite and infinite Beth properties.

Contribution

It proves surjectivity of epimorphisms in many residuated varieties and shows the infinite Beth property is stronger than the finite one, confirming a conjecture.

Findings

Epimorphisms are surjective in all varieties of Heyting or Brouwerian algebras of finite depth.

Surjectivity of epimorphisms holds in varieties of G"odel, Stone, and Sugihara algebras.

In some locally finite varieties of width 2, epimorphisms are not surjective.

Abstract

It is proved that epimorphisms are surjective in a range of varieties of residuated structures, including all varieties of Heyting or Brouwerian algebras of finite depth, and all varieties consisting of Goedel algebras, relative Stone algebras, Sugihara monoids or positive Sugihara monoids. This establishes the infinite deductive Beth definability property for a corresponding range of substructural logics. On the other hand, it is shown that epimorphisms need not be surjective in a locally finite variety of Heyting or Brouwerian algebras of width 2. It follows that the infinite Beth property is strictly stronger than the so-called finite Beth property, confirming a conjecture of Blok and Hoogland.