A new branch of bialgebraic structures: UP-bialgebras

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).

The Third Isomorphism Theorem on UP-Bialgebras

“…Algebraic bi-strukture was analyzed by Vasantha Kandasamy in 2003 [7]. The concept of UP-bialgebras introduced and the first results ware given by Mosrijai and Iampan at the beginning of 2019 [8]. Using by the concept of UP-bihomorphisms, introduced in [8], in this article we formulated and proved the theorem (Theorem 3.3) , which can be viewed as the Third isomorphism theorem between the UP-bialgebras.…”

“…The concept of UP-bialgebras introduced and the first results ware given by Mosrijai and Iampan at the beginning of 2019 [8]. Using by the concept of UP-bihomorphisms, introduced in [8], in this article we formulated and proved the theorem (Theorem 3.3) , which can be viewed as the Third isomorphism theorem between the UP-bialgebras. Of course, there remains an open possibility of formulating and trying to prove other forms of these two isomorphism theorems between the UP-bialgebra.…”

“…In this section, we will present the necessary previous concepts of UP-algebras, their substructures and UP-homomorphisms taken from texts [1,2,3,8]. We will also expose their mutual relationships in the form of proclaims necessary for our intention.…”

“…The concept of UP-bialgebras and some their substructures were introduced and analyzed by Mosrijai and Iampan in the recently published work [8]. In this subsection, taking into account their determinations, we describe the concept of UP-bialgebras and some notions connected with them.…”

“…The concept of bi-algebraic structures was studied by Vasantha Kandasamy in 2003 [7]. The concept of UP-bialgebras with the associated substructures and their mutual connections can be found in [8]. In the forthcoming article [9], this author offered one form the first theorem of the isomorphism between the UP-bialgebras.…”