Discrete duality for 3-valued Łukasiewicz–Moisil algebras

Abstract

In 2011, Düntsch and Orłowska 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 [Formula: see text]-valued Łukasiewicz–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 [Formula: see text] endowed with a unary operation [Formula: see text], satisfying the properties: [Formula: see text] [Formula: see text] and [Formula: see text] 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. Orłowska 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].