Equivalence between Varieties of Łukasiewicz–Moisil Algebras and Rings

Abstract

Abstract The Post, axled and Łukasiewicz–Moisil algebras are important lattices studied in algebraic logic. In this paper, we investigate a useful interpretation between these algebras and some rings. We give a term equivalence between Post algebras of order $p$ and $p$-rings, $p$ prime and lift this result to the axled Łukasiewicz–Moisil algebra $L \cong B_s \times P$ and the ring $\prod ^s F_2 \times \prod ^l F_p$, where $B_s$ is a Boolean algebra of order $2^s$, $P$ a $p$-valued Post algebra of order $p^l$ and $F_p$ is the prime field of order $p$.