Asia Mathematika, Vol 10, issue 1, pages: 1--11.

On the possibility of IF structures

T. Witczak
Institute of Mathematics, University of Silesia, Bankowa 14, 40-007, Katowice, Poland.

Received: 10 Jan 2026 | Accepted: 04 May 2026 | Final Version: 30 May 2026

Abstract

In this paper we investigate the idea of IF pseudo-intuitionistic sets. We combine two possible approaches to the problem of division of the initial universe of discourse into three mutually disjoint parts.
First, we use ideals to model "small" (possibly empty) intersection between the ranges of accepted and rejected objects. In this way we generalize \c{C}oker's intuitionistic sets (implementing the same algebraic operations of union, intersection and complement).
Second, we use filters to describe the fact that the union of both components is "big" (but not necessarily identical with the whole universe). Thus, our structures consist of a non-empty universe $X$ together with some fixed ideal and fixed filter (on $X$). In this setting, we introduce our IF pseudo-intuitionistic sets and we analyze their algebraic properties. We implement the notion of point and we propose several possible directions of future research.

Keywords:
Possibly paraconsistent sets, intuitionistic sets, neutrosophic crisp sets, ideals, filters, IF pseudo-intuitionistic sets.

1. Introduction

We introduced \emph{possibly paraconsistent sets} in [1]. We showed that they are isomorphic with so-called intuitionistic sets (in the sense of coker) and weak rough sets (known as double or flou sets too). Hence, they form an appropriate description of the situation in which our initial universe of discourse is divided into three mutually disjoint parts.

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