Cn algebras with Moisil possibility operators

Abstract

In this paper, we continue the study of the Łukasiewicz residuation algebras of order $n$ with Moisil possibility operators (or $MC_n$-algebras) initiated by Figallo (1989, PhD Thesis, Universidad Nacional del Sur). More precisely, among other things, a method to determine the number of elements of the $MC_n$-algebra with a finite set of free generators is described. Applying this method, we find again the results obtained by Iturrioz and Monteiro (1966, Rev. Union Mat. Argent., 22, 146) and by Figallo (1990, Rep. Math. Logic, 24, 3–16) for the case of Tarski algebras and $I\varDelta _{3}$-algebras, respectively.

Super-Łukasiewicz logics expanded by $Δ$

“…Proof. It follows from (M L7), (M L8), (M L16), and and taking into account the proof given in [24,Theorem 2.11], that h is an homomorphism which verifies that h −1 ({1}) = M . Now, from the first isomorphism theorem, we have there is a one-to-one homomorphism from A/M into L ∆ n as desired.…”

“…Proof. It follows from (M L1) to (M L6) and taking into account the proof given in [24,Lemma 2.4]. ✷ Theorem 2.20 Let A,, ∆, 1 be an LR ∆ n -algebra and let M be a maximal deductive system of A.…”

“…As it said above, in [24], it was studied the n-valued Lukasiewicz residuation algebras expanded with Moisil possibility operators. Taking into account Remark 2.8, it is clear that each LR ∆ nalgebras can be seen as a n-valued Lukasiewicz residuation algebras expanded with Moisil possibility operators.…”