Algebraic structures related to nilpotent minimum algebras and rough sets

Equipping NM-algebras with a Brouwer-like negation ∼, we introduce a kind of algebra under the name BZNMalgebra in the present paper and investigate its algebraic properties in detail. By means of the Brouwer-like negation ∼ and the internal Kleene negation ¬ in NM-algebras, two types of modal operators μ and ν on BZNM-algebras are defined. For any element a of BZNM-algebra, μ(a) and ν(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 is also shown that these two modal operators turn to have an S5 behavior. Additionally, various types of sharp elements with respect to the negation operators including ∼, ¬, (an anti-intuitionistic negation) are defined, and the relationship between these sets and that of ⊗-idempotent (⊕-idempotent) elements are also investigated. It is then proved that the proposed BZNM-algebras are indeed equivalent to NM algebras under a suitable transformation of operators. Furthermore, a special kind of BZNM-algebras, i.e., BZNM3-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 μ and ν. Finally, two interesting examples of BZNM-algebras