Classical structure of rough set theory was first formulated by Z. Pawlak in [6]. The foundation of its object classification is an equivalence binary relation and equivalence classes. The upper and lower approximation operations are two core notions in rough set theory. They can also be seenas a closure operator and an interior operator of the topology induced by an equivalence relation on a u...

Algebraic structures in which the property of commutativity is substituted by mediality are introduced. We consider (associative) graded algebras and instead almost (generalized or $\varepsilon$-commutativity) we introduce ("commutativity-to-mediality" ansatz). Higher twisted products "deforming" brackets (being medial analog Lie brackets) defined. Toyoda's theorem connects (universal) with abe...

Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ elements $x_1,\ldots,x_n$ are equal to each other. The $\mathcal{F}^n_1$ said quasitrivial and those $\mathcal{F}^n_n$ idempotent. We show that $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\subseteq\mathc...