Algebraic structures related to nilpotent minimum algebras and rough sets
Yanhong She, Xiaoli He · Journal of Intelligent & Fuzzy Systems · 2015
Abstract Equipping NM-algebras with a Brouwer-like negation ∼, 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 ∼ 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 S 5 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., 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 μ and ν . Finally, two interesting examples of BZNM-algebras are presented.