Structurally complete finitary extensions of positive \L ukasiewicz logic

TL;DR

This paper characterizes all finitary extensions of the positive fragment of ukasiewicz logic that are structurally complete or hereditarily structurally complete, providing examples and a comprehensive classification.

Contribution

It offers a complete description of structurally complete and hereditarily structurally complete finitary extensions of ukasiewicz positive logic, filling a gap in the logical algebraic theory.

Findings

All finitary structurally complete extensions are described.

Examples of hereditarily structurally complete extensions are provided.

Non-hereditarily structurally complete extensions are also characterized.

Abstract

In this paper we study $\mathcal{MV}^+$, i.e. the positive fragment of {\L}ukasiewicz Multi-Valued Logic $\mathcal{MV}$. In particular we describe all the finitary extensions of $\mathcal{MV}^+$ that are structurally complete and all the axiomatic extensions of $\mathcal{MV}^+$ that are hereditarily structurally complete. Examples of hereditarily structurally complete finitary extensions and non hereditarily structurally complete finitary extensions are provided.