Some Aspects of d‐Units in d/BCK‐Algebras

Abstract

We explore properties of the set of d-units of a d-algebra. A property of interest in the study of d-units in d-algebras is the weak associative property. It is noted that many other d-algebras, especially BCK-algebras, are in fact weakly associative. The existence of d/BCK-algebras which are not weakly associative is demonstrated. Moreover, the notions of a d-integral domain and a left-injectivity are discussed.

Soft BCL-algebras

“…We start by defining a BCL-algebra. We refer the reader to [8], [5] and [7] for the definitions of d-algebra, BCK/BCI-algebra and BCH-algebra respectively. In what follows, is an initial universe set and is a set of parameters.…”

Some Special Elements and Pseudo Inverse Functions in Groupoids

“…The notion of d-algebras, which is another useful generalization of BCK-algebras, was introduced by Neggers and Kim [8], and some relations between d-algebras and BCK-algebras as well as several other relations between d-algebras and oriented digraphs were investigated. Several aspects on d-algebras were studied [9][10][11][12][13][14]. Kim and Neggers [15] introduced the notion of Bin(X), which is the collection of all groupoids defined on a set X, and showed that it becomes a semigroup under suitable operation.…”

Fuzzy Upper Bounds in Groupoids

“…The main assertion is that the squared algebra ( X ; □, 0) of a d -algebra is a d -algebra if and only if the root ( X ; ∗, 0) of the squared algebra ( X ; □, 0) is a strong d -algebra. Recently, Kim et al [5] explored properties of the set of d -units of a d -algebra. It was noted that many d -algebras are weakly associative, and the existence of nonweakly associative d /BCK-algebras was demonstrated.…”