@article { 5667,
title = {On the expressive power of Lukasiewiczâ€™s square operator},
journal = {Journal of Logic and Computation},
volume = {32},
number = {4},
pages = {767-807},
month = {10},
year = {2022},
abstract = {The aim of the paper is to analyze the expressive power of the square operator of \{\L\}ukasiewicz logic: $\ast x=x\odot x$, where $\odot$ is the strong \{\L\}ukasiewicz 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 \{\L\}ukasiewicz 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 \{\L\}ukasiewicz's square operator. Finally, we propose an alternative way to account for \{\L\}ukasiewicz's 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. },
doi={https://doi.org/10.1093/logcom/exab064},
author = {Marcelo Coniglio and Francesc Esteva and Tommaso Flaminio and Llu{\'\i}s Godo} }