On the Variety of m-generalized Łukasiewicz Algebras of Order n

Abstract

In this paper we pursue the study of the variety ${\mathcal{L}^{m}_{n}}$ of m-generalized Łukasiewicz algebras of order n which was initiated in [1]. This variety contains the variety of Łukasiewicz algebras of order n. Given ${{\bf A}\in\mathcal{L}^{m}_{n}}$ , we establish an isomorphism from its congruence lattice to the lattice of Stone filters of a certain Łukasiewicz algebra of order n and for each congruence on A we find a description via the corresponding Stone filter. We characterize the principal congruences on A via Stone filters. In doing so, we obtain a polynomial equation which defines the principal congruences on the algebras of ${\mathcal{L}^{m}_{n}}$ . After showing that for m > 1 and n > 2, the variety of Łukasiewicz algebras of order n is a proper subvariety of ${\mathcal{L}^{m}_{n}}$ , we prove that ${\mathcal{L}^{m}_{n}}$ is a finitely generated discriminator variety and point out some consequences of this strong property, one of which is congruence permutability.