Inf-structuring Functions and Self-dual Marked Flattenings in bi-Heyting Algebra

Abstract

This paper introduces a generalization of self-dual marked flattenings defined in the lattice of mappings. This definition provides a way to associate a self-dual operator to every mapping that decomposes an element into sub-elements (i.e. gives a cover). Contrary to classical flattenings whose definition relies on the complemented structure of the powerset lattices, our approach uses the pseudo relative complement and supplement of the bi-Heyting algebra and a new notion of inf-structuring functions that provides a very general way to structure the space. We show that using an inf-structuring function based on connections allows to recover the original definition of marked flattenings and we provide, as an example, a simple inf-structuring function whose derived self-dual operator better preserves contrasts and does not introduce new pixel values.