Discrete duality for 3-valued Lukasiewicz-Moisil algebras

Abstract

In 2011, Düntsch and Orlowska obtained a discrete duality for regular double Stone algebras. On the other hand, it is well known that regular double Stone algebras are polinominally equivalent to 3-valued Lukasiewicz-Moisil algebras (or LM3-algebras). In [R. Cignoli, Injective De Morgan and Kleene algebra, Proc. Amer. Math. Soc. 47 (1975) 269-278], LM3-algebras are considered as a Kleene algebras (L,∨,∧,∼, 0, 1) endowed with a unary operation : L → L, satisfying the properties: a∨ ∼ a = 1, ∼ a ∧ a = a∧ ∼ a and a∨b ≤(a ∨ b). Motivated by this result, in this paper, we determine another discrete duality for LM3-algebras, extending the discrete duality to De Morgan algebras described in [W. Dzik, E. Orlowska and C. van Alten, Relational representation theorems for general lattices with negations, in Relations and Kleene Algebra in Computer Science, Lecture Notes in Computer Science, Vol. 4136 (Springer, Berlin, 2006), pp. 162-176].