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Algebraic structures related to nilpotent minimum algebras and rough sets. (English) Zbl 1364.03090
Summary: Equipping NM-algebras with a Brouwer-like negation $$\sim$$, we introduce a kind of algebra under the name BZNM-algebra in the present paper and investigate its algebraic properties in detail. By means of the Brouwer-like negation $$\sim$$ and the internal Kleene negation $$\neg$$ in NM-algebras, two types of modal operators $$\mu$$ and $$\nu$$ on BZNM-algebras are defined. For any element $$a$$ of BZNM-algebra, $$\mu(a)$$ and $$\nu(a)$$ turn out to be the best approximation of $$a$$ from the bottom and the top, respectively, with Boolean skeleton serving as the collection of sharp elements. It isalso shown that these two modal operators turn to have an $$S_5$$ behavior. Additionally, various types of sharp elements with respect to the negation operators including $$\sim$$, $$\neg$$, $$\flat$$, (an anti-intuitionistic negation) are defined, and the relationship between these sets and that of $$\otimes$$-idempotent ($$\oplus$$-idempotent) elements are also investigated. It is then proved that the proposed BZNM-algebras are indeed equivalent to NMalgebras under a suitable transformation of operators. Furthermore, a special kind of BZNM-algebras, i.e., BZNM$$^3$$-algebras, is investigated and its equivalent characterizations are given. A new type of rough approximation operators are proposed, and it is shown that such a pair of approximation operators is equivalent to the defined operators $$\mu$$ and $$\nu$$. Finally, two interesting examples of BZNM-algebras are presented.

##### MSC:
 03G25 Other algebras related to logic 06D35 MV-algebras
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