On \(n \times m\)-valued Łukasiewicz-Moisil algebras
\(n\times m\)-valued \(\L\)ukasiewicz algebras with negation (called \(\text{LM}_{n\times m}\)-algebras) were introduced and investigated by the author of this article and constitute a non-trivial generalization of \(n\)-valued \(\L\)ukasiewicz-Moisil algebras. This paper continues the study of these algebras, more precisely, the author describes a topological duality for these algebras (the dual of the category of \(\text{LM}_{n\times m}\)-algebras is the category \(\mathbf{lm}_{n\times m}\) of \(\text{lm}_{n\times m}\)-spaces and \(\text{lm}_{n\times m}\)-functions) and characterizes the congruences on these algebras by means of special subsets of the associated spaces. In particular, the author determines which of these subsets correspond to principal congruences. Also, the author proves that the variety of \(n\times m\)-valued \(\L\)ukasiewicz-Moisil algebras is a discriminator variety (if it has a polynomial \(p\) that coincides with the ternary discriminator function on each subdirectly irreducible member of the variety) and, as a consequence, she obtains certain properties of the congruences, such as: every principal congruence is a factor congruence, principal congruences form a sublattice in the congruence lattice, each compact congruence is a principal congruence, the congruences are regular, normal, filtral, and the category \(\mathbf{LM}_{n\times m}\) has the congruence extension property. Finally, she computes the number of congruences of such finite algebras: if \(L\) is a finite \(\text{LM}_{n\times m}\)-algebra and \(A\) is the set of all atoms of its center, then \(|\text{Con}_{\text{LM}_{n\times m}}(L)| =2^{A}\).
scientific article; zbMATH DE number 5343479
scientific article