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.

Determining the number of discrete operators has been a topic of interest for the scientific community since the introduction of these operators. This paper represents a further stage within this topic in the field of operators defined on a finite chain. Mainly, two families of discrete operators are studied: discrete conjunctions that are smooth (a property that is usually considered the equivalent to continuity in discrete settings) and commutative discrete conjunctions with n, the greatest element of the chain, as neutral element, so that only associativity is missing to become discrete t-norms. To determine the cardinality of the first family, we study its explicit representation by alternating sign matrices, obtaining that the cardinality is preserved between both structures and allowing us to relate intrinsic properties of the family of discrete operators with intrinsic properties of such class of matrices. For commutative discrete conjunctions with n as neutral element, we have considered the concepts of n-Gog y n-Magog triangles, allowing us to transform properties of these operators into properties of these triangular arrays; in particular, their cardinality. In this way, an upper bound for the cardinality of discrete t-norms is achieved.

Many fields in applied sciences, like Artificial Intelligence and Computer Science, use aggregation methods to provide new generalized metrics from a collection of old ones. Thus, the problem of merging by means of a function a collection of generalized metrics into a single one has been recently studied in depth. Moreover, the mipoint sets for a generalized metric involving fuzzy sets have shown a great potential in medical diagnosis and decision making since it models the concept of “compromise” or “middle way” between two positions. Joining these facts, the aim of this paper is to provide a general framework for the study of midpoint sets for quasimetrics via aggregation theory. In particular, we determine the properties that an aggregation function must satisfy to characterize the midpoint set for a quasimetric generated by means of the fusion of a collection of quasimetrics in terms of the midpoint sets for each of the quasimetrics that are merged. In fact, this study generalizes the results for metrics in this context that are retrieved as a particular case of the exposed theory. Finally, some particular results for generalized metrics defined for fuzzy sets are proved.