n x m Łukasiewicz algebras with parameters

Matrix Łukasiewicz algebras were introduced by W. Suchoń in 1975 (Matrix Łukasiewicz Algebras, Reports on Mathematical Logic 4 (1975), 91-104). In this paper n x m{valued Łukasiewicz algebras with negation (or NSnxm algebras are defined and investigated. These algebras constitute an extension of those given by W. Suchoń and in m = 2 case they coincide with n valued Łukasiewicz algebras. Firstly, some of the main results established for matrix Łukasiewicz algebras are extended to NSnxm algebras. In particular, a functional representation theorem is given. Next, NSnxm congruences are determined by taking into account an implication operation which is defined on these algebras. In addition, it is proved that the class of NSnxm algebras is a variety. Besides, subdirectly irreducible algebras are characterized. As a consequence, it is shown that this variety is semisimple and locally finite. Finally, the algebra which generates the variety of NSnxm algebras is obtained and an equational base for the latter is determined.