# Selfextensional logics with a distributive nearlattice term

## TL;DR

This paper introduces distributive nearlattice terms in algebraic logic, characterizes selfextensional logics with such terms, and proves their associated algebraic classes form varieties, ensuring full selfextensionality.

## Contribution

It defines DN-terms within algebraic languages, characterizes selfextensional logics with DN-terms, and establishes their algebraic classes as varieties, advancing the understanding of algebraic semantics.

## Key findings

- Distributive nearlattice terms generalize Tarski algebras and lattices.
- Selfextensional logics with DN-terms are characterized via algebraic interpretation.
- The algebraic classes associated with these logics form varieties, confirming full selfextensionality.

## Abstract

We define when a ternary term $m$ of an algebraic language $\mathcal{L}$ is called a \textit{distributive nearlattice term} (DN-term) of a sentential logic $\mathcal{S}$. Distributive nearlattices are ternary algebras generalising Tarski algebras and distributive lattices. We characterise the selfextensional logics with a DN-term through the interpretation of the DN-term in the algebras of the algebraic counterpart of the logics. We prove that the canonical class of algebras (under the point of view of Abstract Algebraic Logic) associated with a selfextensional logic with a DN-term is a variety, and we obtain that the logic is in fact fully selfextensional.