Abstract

In 2004, C. Sanza, with the purpose of legitimizing the study of \-valued Łukasiewicz algebras with negation -algebras) introduced \-valued Łukasiewicz algebras with negation. Despite the various results obtained about \-algebras, the structure of the free algebras for this variety has not been determined yet. She only obtained a bound for their cardinal number with a finite number of free generators. In this note we describe the structure of the free finitely generated \-algebras and we determine a formula to calculate its cardinal number in terms of the number of free generators. Moreover, we obtain the lattice \\) of all subvarieties of \ and we show that the varieties of Boolean algebras, three-valued Łukasiewicz algebras and four-valued Łukasiewicz algebras are proper subvarieties of \.