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

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).