N-SUBALGEBRAS OF TYPE (∈, ∈ ∨ q) BASED ON POINT N-STRUCTURES IN BCK/BCI-ALGEBRAS

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Abstract. CharacterizationsofN-subalgebraoftype(∈,∈∨q)arepro-vided. ThenotionofN-subalgebrasoftype(∈,∈∨q)isintroduced,andits characterizations are discussed. Conditions for an N-subalgebra oftype(∈,∈∨q)(resp. (∈,∈∨q))tobeanN-subalgebraoftype(∈,∈)areconsidered. 1. IntroductionA (crisp) set A in a universe X can be defined in the form of its character-istic function µ A : X → {0,1} yielding the value 1 for elements belonging tothe set A and the value 0 for elements excluded from the set A. So far mostof the generalization of the crisp set have been conducted on the unit inter-val [0,1] and they are consistent with the asymmetry observation. In otherwords, the generalization of the crisp set to fuzzy sets relied on spreading pos-itive information that fit the crisp point {1} into the interval [0,1]. Because nonegative meaning of information is suggested, we now feel a need to deal withnegative information. To do so, we also feel a need to supply mathematicaltool. To attain such object, Jun et al. [5] introduced a new function whichis called negative-valued function, and constructed N-structures. They ap-plied N-structures to BCK/BCI-algebras, and discussed N-subalgebras andN-ideals in BCK/BCI-algebras. Jun et al. [6] considered closed ideals inBCH-algebras based on N-structures. Also, using N-structures, Jun and Leeintroduced the notion of an N-essence in a subtraction algebra, and inves-tigated related properties. They discussed relations among an N-ideal, anN-subalgebra and an N-essence (see [4]). To obtain more general form of anN-subalgebra in BCK/BCI-algebras, Jun et al. defined the notions of N-subalgebras of types (∈,∈), (∈, q), (∈,∈ ∨q), (q,∈), (q, q) and (q,∈ ∨q),and investigated related properties. They provided a characterization of an N-subalgebra of type (∈,∈∨q). They also gave conditions for an N-structure to