On the expressive power of Lukasiewicz's square operator

TL;DR

This paper investigates the expressive capabilities of Lukasiewicz's square operator, characterizing when it can reconstruct finite MV-chains and axiomatizing related algebraic logics, with applications to G"odel chains.

Contribution

It provides a complete characterization of when the square operator reconstructs MV-chains, axiomatizes the associated logic, and offers an alternative interpretation on G"odel chains.

Findings

Reconstruction of MV-chains is possible only for chains of prime power order.

A new axiomatization of the algebraizable matrix logic is provided.

An intuitive set of equations captures the operator on G"odel chains.

Abstract

The aim of the paper is to analyze the expressive power of the square operator of Lukasiewicz logic: $\ast x=x\odot x$, where $\odot$ is the strong Lukasiewicz conjunction. In particular, we aim at understanding and characterizing those cases in which the square operator is enough to construct a finite MV-chain from a finite totally ordered set endowed with an involutive negation. The first of our main results shows that, indeed, the whole structure of MV-chain can be reconstructed from the involution and the Lukasiewicz square if and only if the obtained structure has only trivial subalgebras and, equivalently, if and only if the cardinality of the starting chain is of the form $n+1$ where $n$ belongs to a class of prime numbers that we fully characterize. Secondly, we axiomatize the algebraizable matrix logic whose semantics is given by the variety generated by a finite totally ordered set endowed with an involutive negation and Lukasiewicz's square operator. Finally, we propose an alternative way to account for Lukasiewicz square operator on involutive G\"odel chains. In this setting, we show that such an operator can be captured by a rather intuitive set of equations.