An algebraic construction of Moisil operators in (n + 1)-valued {\L}ukasiewicz propositional calculus

MARTÍN FIGALLO; ALDO VICTORIO FIGALLO

Journal of Multiple-Valued Logic and Soft Computing

OLD CITY PUBLISHING INC

Año: 2013 vol. 21 p. 131 - 145

1542-3980

{L}ukasiewcz residuation algebras of order $(n+1)$ (or C$_{n+1}$-algebras) with Moisil possibility operators were introduced by A. V. Figallo in cite{AF1}. These algebras constitute a variety and are algebraic models of a fragment of the $(n+1)$-valued {L}ukasiewicz propositional calculus. In this calculus {L}ukasiewicz implication $ ightarrow$ along with unary conectives $(sigma_i)_{i in J}$, better known as Moisil possibility operators, are taken as primitives. Let $L_{n+1}={0, rac{1}{n}, rac{2}{n}, dots, rac{n-1}{n},1 }$ and $J={1,2, dots, n}$. Then, this variety is generated by the algebra $langle L_{n+1}, ightarrow , (sigma_i)_{iin J}, 1 angle$, where $ ightarrow$ is defined by $x ightarrow y=mbox{ min} {1,1-x+y}$ and $sigma_i:L_{n+1}longrightarrow L_{n+1}$ is defined by egin{equation*}label{eqabstract} sigma _i left(rac{j}{n} ight)=left{ egin{array}{ll} 0 & mbox{if $j + i < n+1 $} [4mm] 1 & mbox{if $j + igeq n +1 $} end{array} ight. , mbox{ for every } jin {0}cup J, i in J. end{equation*} oindent In the present work, we describe a method, inspired by the one developed by W. Sucho´n in cite{WS}, for constructing unary operators $sigma_i$, $2leq i leq n$, from $ ightarrow$ and $sigma_1$.