Ambiguous representations of semilattices and imperfect information

TL;DR

This paper introduces ambiguous representations of continuous semilattices, explores their categorical properties, and discusses applications to processing imperfect information.

Contribution

It defines ambiguous representations and pseudo-inverse operations, establishing categorical structures and dualities for semilattices with applications to imperfect information.

Findings

Ambiguous representations form categories with involutive pseudo-inverses.

Self-dualities and contravariant equivalences are established.

Applications to imperfect information processing are discussed.

Abstract

Crisp and lattice-valued ambiguous representations of one continuous semilattice in another one are introduced and operation of taking pseudo-inverse of the above relations is defined. It is shown that continuous semilattices and their ambiguous representations, for which taking pseudo-inverse is involutive, form categories. Self-dualities and contravariant equivalences for these categories are obtained. Possible interpretations and applications to processing of imperfect information are discussed.