On the action of the implicative closure operator on the set of partial functions of the multivalued logic

Abstract

AbstractOn the setPk∗$\begin{array}{} \displaystyle P_k^* \end{array}$of partial functions of thek-valued logic, we consider the implicative closure operator, which is the extension of the parametric closure operator via the logical implication. It is proved that, for anyk⩾ 2, the number of implicative closed classes inPk∗$\begin{array}{} \displaystyle P_k^* \end{array}$is finite. For anyk⩾ 2, inPk∗$\begin{array}{} \displaystyle P_k^* \end{array}$two series of implicative closed classes are defined. We show that these two series exhaust all implicative precomplete classes. We also identify all 8 atoms of the lattice of implicative closed classes inP3∗$\begin{array}{} \displaystyle P_3^* \end{array}$.