Transactions on Rough Sets XXII

In this thesis, a generalization of the classical Rough set theory [83] is developed considering the so-called sequences of orthopairs that we define in [20] as special sequences of rough sets.

Mainly, our aim is to introduce some operations between sequences of orthopairs, and to discover how to generate them starting from the operations concerning standard rough sets (defined in [32]). Also, we prove several representation theorems representing the class of finite centered Kleene algebras with the interpolation property [31], and some classes of finite residuated lattices (more precisely, we consider Nelson algebras [87], Nelson lattices [23], IUML-algebras [73] and Kleene lattice with implication [27]) as sequences of orthopairs.

Moreover, as an application, we show that a sequence of orthopairs can be used to represent an examiner’s opinion on a number of candidates applying for a job, and we show that opinions of two or more examiners can be combined using operations between sequences of orthopairs in order to get a final decision on each candidate.

Finally, we provide the original modal logic \(SO_n\) with semantics based on sequences of orthopairs, and we employ it to describe the knowledge of an agent that increases over time, as new information is provided. Modal logic \(SO_n\) is characterized by the sequences \((\square _1, \ldots , \square _n)\) and \((\bigcirc _1, \ldots , \bigcirc _n)\) of n modal operators corresponding to a sequence \((t_1, \ldots , t_n)\) of consecutive times. Furthermore, the operator \(\square _i\) of \((\square _1, \ldots , \square _n)\) represents the knowledge of an agent at time \(t_i\), and it coincides with the necessity modal operator of S5 logic [29]. On the other hand, the main innovative aspect of modal logic \(SO_n\) is the presence of the sequence \((\bigcirc _1, \ldots , \bigcirc _n)\), since \(\bigcirc _i\) establishes whether an agent is interested in knowing a given fact at time \(t_i\).

The seminal work of Z. Pawlak [60] on rough set theory has attracted the attention of researchers from various disciplines. Algebraists introduced some new algebraic structures and represented some old existing algebraic structures in terms of algebras formed by rough sets. In Logic, the rough set theory serves the models of several logics. This paper is an amalgamation of algebras and logics of rough set theory.

We prove a structural theorem for Kleene algebras, showing that an element of a Kleene algebra can be looked upon as a rough set in some appropriate approximation space. The proposed propositional logic \(\mathcal {L}_{K}\) of Kleene algebras is sound and complete with respect to a 3-valued and a rough set semantics.

This article also investigates some negation operators in classical rough set theory, using Dunn’s approach. We investigate the semantics of the Stone negation in perp frames, that of dual Stone negation in exhaustive frames, and that of Stone and dual Stone negations with the regularity property in \(K_{-}\) frames. The study leads to new semantics for the logics corresponding to the classes of Stone algebras, dual Stone algebras, and regular double Stone algebras. As the perp semantics provides a Kripke type semantics for logics with negations, exploiting this feature, we obtain duality results for several classes of algebras and corresponding frames.

In another part of this article, we propose a granule-based generalization of rough set theory. We obtain representations of distributive lattices (with operators) and Heyting algebras (with operators). Moreover, various negations appear from this generalized rough set theory and achieved new positions in Dunn’s Kite of negations.