Simple and Subdirectly Irreducible Tense 𝐿𝑀θ-algebras
The main result of this paper is the determination of simple and subdirectly irreducible tense θ-valued Łukasiewicz–Moisil algebras (or tense 𝐿𝑀θ-algebras). To reach this aim, we use the topological duality for these algebras obtained by Figallo, Pascual and Pelaitay in 2019 and also the filters of these algebras. Besides, we define θ-simple and θ-subdirectly irreducible tense 𝐿𝑀θ-algebras and we describe them. Moreover, we establish a relationship between simple and subdirectly irreducible tense 𝐿𝑀θ-algebras and θ-simple and θ-subdirectly irreducible tense 𝐿𝑀θ-algebras, respectively. Finally, we get a characterization of the subdirectly irreducible algebras which are complete and completely chrysippian tense 𝐿𝑀θ-algebras and whose θ-filters are complete.
Keywords: Tense θ-valued Łukasiewicz–Moisil algebras, topological duality, congruences, θ-congruences, simple and subdirectly irreducible algebras