Free algebras in varieties of Hilbert algebras with supremum generated by finite chains

TL;DR

This paper studies free algebras in varieties of n-valued Hilbert algebras with supremum, generated by finite chains, and determines bounds on their cardinality, advancing understanding of their algebraic structure.

Contribution

It introduces and analyzes the class of n-valued Hilbert algebras with supremum, focusing on free algebras generated by finite chains and establishing bounds on their size.

Findings

Varieties Hn are generated by finite chains.

Determined an upper bound for the cardinality of free Hn-algebras.

Analyzed the structure of free algebras with multiple generators.

Abstract

Hilbert algebras with supremum, i.e., Hilbert algebras where the associated order is a join-semilattice were first considered by A.V. Figallo, G. Ramon and S. Saad in [11], and independently by S. Celani and D. Montangie in [7]. On the other hand, L. Monteiro introduced the notion of n-valued Hilbert algebras (see [12]). In this work, we investigate the class of n-valued Hilbert algebras with supremum, denoted Hn, i.e., n-valued Hilbert algebras where the associated order is a join-semilattice. The varieties Hn are generated by finite chains. The free Hn-algebra Freen+1(r) with r generators is studied. In particular, we determine an upper bound to the cardinal of the finitely generated free algebra Freen+1(r).