Induced operators on bounded lattices

Abstract

In this paper we show a methodology for designing operators on spaces of lattice-valued mappings. More precisely, from a family of operators on a bounded lattice L and mappings from a set X to itself, we may construct an operator, that we call the induced operator, on the lattice of set mappings from X to L. Furthermore, if X is also a bounded lattice, under suitable conditions preserving the orders on L and X, the induced operator belongs to the lattice of monotone mappings from X to L. The procedure is quite simple, versatile and allows to obtain plenty of different examples in a wide range of lattices. In particular, by appropriate choices of X and L, it can be applied to the most important types of fuzzy sets. The relation with some properties associated to popular types of operators is studied. Hence, we show that, under certain conditions, aggregation operators, implications, negations, overlap functions and others are preserved by the induction process.