Characterizations of regular ordered semigroups in terms of (α, β)-bipolar fuzzy generalized bi-ideals

Regular Ordered Ternary Semigroups in Terms of Bipolar Fuzzy Ideals

“…We have proved that if a bipolar fuzzy set in an ordered ternary semigroup is a (∈, ∈ ∨q)-bipolar fuzzy generalized bi-ideal of S, it satisfies two particular properties of S. However, we have shown with an example that the reverse of this statement is not true in general. This result has negated the result proved in [20].…”

“…The (∈, ∈ ∨q)-bipolar fuzzy generalized bi-ideal has played a central role in the (α, β)-bipolar fuzzy generalized bi-ideal. The characterizations of the regular ordered semigroups in terms of (α, β)-fuzzy generalized bi-ideals and in terms of (α, β)-bipolar fuzzy generalized bi-ideals were given in [3,20]. Here, we extend their generalizations to ordered ternary semigroups.…”

“…We have characterized the regular ordered ternary semigroups in terms of the bipolar fuzzy ideals and (α, β)-bipolar fuzzy generalized bi-ideals. Several results in [20,21] are generalized and new results are found.…”

“…Lee [19] presented the notion of bipolar fuzzy ideals of BCK\BCI-algebras. Ibrar, Khan and Davvaz [20] used (α, β)-bipolar fuzzy generalized bi-ideals for the characterizations of the regular ordered semigroups. In this paper, we have proved that a set of bipolar fuzzy subsets in an ordered ternary semigroup is also an ordered ternary semigroup.…”

“…In this paper, we have proved that a set of bipolar fuzzy subsets in an ordered ternary semigroup is also an ordered ternary semigroup. Before this paper, this theorem was just used for ordered semigroups in [20]. Similarly, we have outlined our next major contribution of this paper.…”