Question

what is highper congrugence​

Answer 1

Explanation:

REGULAR CONGRUENCE RELATIONS ON HYPER BCK-ALGEBRAS

R. A. BORZOOEI, H. HARIZAVI

Received September 5, 2003; revised June 23, 2004

Abstract. In this manuscript first by definition of regular congruence relation on a

hyper BCK-algebra, we construct a quotient hyper BCK-algebra. After that, we state

and prove the homomorphism and isomorphism theorems for hyper BCK-algebras. Fi-

nally, we show that there exists at least one maximal regular congruence relation in a

bounded hyper BCK-algebra.

1. Introduction

The study of BCK-algebras was initiated by Y. Imai and K. Is´eki[5] in 1966 as a gen-

eralization of the concept of set-theoretic difference and propositional calculi. Since then a

great deal of literature has been produced on the theory of BCK-algebras. In particular,

emphasis seems to have been put on the ideal theory of BCK-algebras. The hyperstruc-

ture theory (called also multialgebras)was introduced in 1934 by F. Marty [10] at the 8th

congress of Scandinavian Mathematiciens. Around the 40’s, several authors worked on

hypergroups, especially in France and in the United States, but also in Italy, Russia and

Japan. Over the following decades, many important results appeared, but above all since

the 70’s onwards the most luxuriant flourishing of hyperstructures has been seen. Hyper-

structures have many applications to several sectors of both pure and applied sciences. In

[8], Y. B. Jun et al. applied the hyperstructures to BCK-algebras, and introduced the

notion of a hyper BCK-algebra which is a generalization of BCK-algebra, and investigated

some related properties. They also introduced the notions of hyper BCK-ideal, strong

and reflexive hyper BCK-ideals. Now we follow [9] and introduce the concept of quotient

hyper BCK-algebras. Then we prove homomorphism and isomorphism theorems for hyper

BCK-algebras and we get some related results. Finally, we show that there exists at least

one maximal regular congruence relation in a bounded hyper BCK-algebra.

2. Preliminaries

Definition 2.1. [8] By a hyper BCK-algebra we mean a non-empty set H endowed with

a hyperoperation “◦” and a constant 0 satisfying the following axioms:

(HK1) (x ◦ z) ◦ (y ◦ z) -

x ◦ y,

(HK2) (x ◦ y) ◦ z = (x ◦ z) ◦ y,

(HK3) x ◦ H -

{x},

(HK4) x -

y and y -

x imply x = y.

for all x, y, z ∈ H, where x -

y is defined by 0 ∈ x ◦ y and for every A, B ⊆ H, A -

B is

defined by ∀a ∈ A, ∃b ∈ B such that a -

b. In such case, we call “-

” the hyperorder in H.