Abstract
The notion of bialgebraic structures was discussed by Vasantha Kandasamy [Bialgebraic structures and Smarandache bialgebraic structures. India: American Research Press; 2003]. The main target of this paper is to introduce the notions of a KU-bialgebra, a KP-bialgebra, a PK-bialgebra, and a UP-bialgebra and the notions of a UP-bisubalgebra, a UP-bifilter, a UP-biideal, and a strongly UP-biideal of UP-bialgebras, and prove the generalization of the notions and some results related to a UP-subalgebra, a UP-filter, a UP-ideal, and a strongly UP-ideal of UP-algebras. Furthermore, we introduce the notion of a UP-bihomomorphism and study the image and inverse image of a UP-bisubalgebra, a UP-bifilter, a UP-biideal, and a strongly UP-biideal of UP-bialgebras under a UP-bihomomorphism. Finally, we have the generalization diagram of KU/KP/PK/UP-bialgebras (see Figure 1) and the diagram of special subsets of UP-bialgebras (see Figure 2).
Highlights
Among many algebraic structures, algebras of logic form important class of algebras
The main target of this paper is to introduce the notions of a KU-bialgebra, a KP-bialgebra, a PK-bialgebra, and a UP-bialgebra and the notions of a UP-bisubalgebra, a UP-bifilter, a UP-biideal, and a strongly UP-biideal of UP-bialgebras, and prove the generalization of the notions and some results related to a UP-subalgebra, a UP-filter, a UP-ideal, and a strongly UP-ideal of UP-algebras
It have been examined by several researchers, for example, the notion of derivations of UP-algebras was introduced by Sawika et al [7], Somjanta et al [8] introduced the notion of fuzzy sets in UP-algebras, the concept of hesitant fuzzy sets on UP-algebras was introduced by Mosrijai et al [9], Senapati et al [10, 11] applied cubic set and interval-valued intuitionistic fuzzy structure in UP-algebras, Romano [12] introduced the notion of proper UP-filters in UPalgebras, Iampan et al [13] introduced the concept of a partial transformation UP-algebra induced by a UP-algebra, etc
Summary
Algebras of logic form important class of algebras Examples of these are BCK-algebras [1], BCI-algebras [2], KU-algebras [3], UP-algebra [4] and others (see [5, 6]). Example 2.6: Let A = {0, 1, 2, 3, 4} be a set with a binary operation · defined by the following Cayley table:. Definition 2.7 ([4, 8, 21]): A nonempty subset S of a UP-algebra (A, ·, 0) is called (1) a UP-subalgebra of A if (∀ x, y ∈ S)(x · y ∈ S). The dually set {x ∈ A | f (x) ∈ D} which denoted by f −1(D) is called the inverse image of D under f. F −1(D) is a strongly UP-ideal of A
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