A study on the cardinality of some families of discrete operators through alternating sign matrices

Abstract

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.