On the cardinality of some families of discrete connectives

Abstract

The computation of a closed formula for the cardinality of some discrete connectives has received the interest of the research community since the beginning of this class of operators. This paper constitutes a substantial progress in this topic. First, monotonicities and other properties of discrete connectives are related to plane partitions, a concept deeply studied in the field of combinatorics. Second, the already known expressions on the cardinality of plane partitions are adapted to the concrete properties of discrete connectives. With this, we establish closed formulas for discrete negations and some binary discrete connectives; concretely, discrete conjunctions, disjunctions and implications. As well, we establish formulas for the cardinality of some sets of implications satisfying several additional properties known in the literature.