2021/03/12 by Marcelo E. Coniglio, Francesc Esteva, Coniglio, Marcelo E. +5
Computer Science · #03G10 #03G20 #06B20 #Advanced Algebra and Logic #FOS: Mathematics #Logic (math.LO) #Logic, Reasoning, and Knowledge #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2103.07548
openalex publication_date 2021/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The aim of the paper is to analyze the expressive power of the square operator of Lukasiewicz logic: ∗ 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 Gödel chains. In this setting, we show that such an operator can be captured by a rather intuitive set of equations.