  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.
