یانگ بای جان

By applying the concept of the Lukasiewicz fuzzy set to the filter in hoops, the Lukasiewicz fuzzy filter is introduced and its properties are investigated.The relationship between fuzzy filter and Lukasiewicz fuzzy filter is discussed.Conditions for the Lukasiewicz fuzzy set to be a Lukasiewicz fuzzy filter are provided, andcharacterizations of Lukasiewicz fuzzy filter are displayed.The conditions under which the three subsets, $\in$-set, q-set, and O-set, will be filter are explored.

The notions of (transitive, commutative, antisymmetric) bordered GE-algebras are introduced,and their properties are investigated. Relations between a commutative bordered GE-algebra and anantisymmetric bordered GE-algebra are considered, and also relations between a commutative borderedGE-algebra and a transitive bordered GE-algebra are discussed. Relations between a bordered GE-algebra and a bounded Hilbert algebra are stated, and the conditions under which every bordered GE-algebra (resp., bounded Hilbert algebra) can be a bounded Hilbert algebra (resp., bordered GE-algebra) are found. The concept of duplex bordered GE-algebras is introduced, and its properties are investigated. Relations between an antisymmetric bordered GE-algebra and a duplex bordered GE-algebra are discussed, and the conditions under which an antisymmetric bordered GE-algebra can be a duplex GE-algebra are established. A characterization of a duplex bordered GE-algebra is provided. A new bordered GE-algebra called cross bordered GE-algebra which is wider than duplex bordered GE-algebra is introduced, and its properties are investigated. Relations between a duplex bordered GE-algebra and a cross bordered GE-algebra are considered.

The notion of fuzzy ideal in bounded equality algebras is defined, and several properties are studied. Fuzzy ideal generated by a fuzzy set is established, and a fuzzy ideal is made by using the collection of ideals. Characterizations of fuzzy ideal are discussed. Conditions for a fuzzy ideal to attains its infimum on all ideals are provided. Homomorphic image and preimage of fuzzy ideal are considered. Finally, quotient structures of equality algebra induced by (fuzzy) ideal are studied.

The notion of commutative falling intuitionistic fuzzy ideal of a BCK-algebra is introduced and related properties are investigated. We verify that every commutative intuitionistic fuzzy ideal is a commutative falling intuitionistic fuzzy ideal, and provide example to show that a commutative falling intuitionistic fuzzy ideal is not a commutative intuitionistic fuzzy ideal. Relations between a falling intuitionistic fuzzy ideal and a commutative falling intuitionistic fuzzy ideal are considered, and a condition for a falling intuitionistic fuzzy ideal to be a commutative falling intuitionistic fuzzy ideal is provided.