A Study of Algebraic Structures and Logics Based on Categories of Rough Sets

The theory of rough sets has been studied extensively, both from foundation and application points of view, since its introduction by Pawlak in 1982. On the foundations side, a substantial part of work on rough set theory involves the study of its algebraic aspects and logics. The present work is in this direction, initiated through the study of categories of rough sets.Starting from two categories RSC and ROUGH of rough sets, it is shown that they are equivalent. Moreover, RSC, and thus ROUGH, are found to be a quasitopos, a structure slightly weaker than topos. The construction is then lifted to a more general set-up to give the category RSC( $$\mathscr {C}$$ C ) with an arbitrary non-degenerate topos $$\mathscr {C}$$ C serving as a ‘base’, just as sets constitute a base for defining rough sets.The category-theoretic study gives rise to two directions of work. In one direction of work, a particular example of RSC( $$\mathscr {C}$$ C ) when $$\mathscr {C}$$ C is the topos of monoid actions on sets is considered. It yields the monoid actions on rough sets and that of transformation semigroups (ts) for rough sets, leading to decomposition results. A semiautomaton for rough sets is also defined.In the other direction, we incorporate Iwinski’s notion of ‘relative rough complementation’ in the internal algebra of the quasitopos RSC( $$\mathscr {C}$$ C ). This results in the introduction of two new classes of algebraic structures with two negations, namely contrapositionally complemented pseudo-Boolean algebra (ccpBa) and contrapositionally $${\vee }$$ ∨ complemented pseudo-Boolean algebra (c $$\vee $$ ∨ cpBa). Examples of ccpBas and c $$\vee $$ ∨ cpBas are developed, comparison with existing algebras is done and representation theorems are established.The logics ILM and ILM- $$\vee $$ ∨ corresponding to ccpBas and c $${\vee }$$ ∨ cpBas respectively are defined, and different relational semantics are obtained. It is shown that ILM is a proper extension of a variant JP $$'$$ ′ of Peirce’s logic, defined by Segerberg in 1968. The inter-relationship between relational semantics and the algebraic semantics of ILM and ILM- $${\vee }$$ ∨ are investigated. Lastly, in the line of Dunn’s study of logics, the two negations are expressed without the help of the connective of implication, and the resulting logical and algebraic structures are also studied.