Uma extensão de overlaps e naBL-Álgebras para reticulados

Abstract

Overlap functions were introduced as a class of bivariate aggregation functions on [0, 1] to be applied in the image processing field. Many researchers have begun to develop overlap functions in order to explore their potential in different scenarios, such as problems involving classification or decision making. Recently, a non-associative generalization of Hájek’s BL-algebras (naBL-algebras) were investigated from the perspective of overlap functions as a residuated application. In this work, we generalize the notion of overlap functions for the lattice context and introduce a weaker definition, called a quasi-overlap, that arises from definition, called a quasi-overlap, that arises from the removal of the continuity condition. To this end, the main properties of (quasi-) overlaps over bounded lattices, namely: convex sum, migrativity, homogeneity, idempotency, and cancellation law are investigated, as well as an overlap characterization of Archimedian overlap functions is presented. In addition, we formalized the residual principle for the case of quasi-overlap functions on lattices and their respective induced implications, as well as revealing that the class of quasi-overlap functions that fulfill the residual principle is the same class of continuous functions according the topology of Scott. As a consequence, we provide a new generalization of the notion of naBL-algebras based on overlap over lattices.