An overview of hyper logical algebras

Abstract

Hyper logical algebras were first studied in 2000 by Borzooei et al. They applied the concept of hyperstructures to one of the logical algebraic structures known as the BCK-algebra, and introduced two generalizations of them called the hyper BCK-algebra and hyper K-algebra. Then many researchers in this field continued their research and used hyperstructures on other logical algebras and introduced the concepts of hyper residuated lattices, hyper BL-algebras, hyper MV-algebras, hyper EQ-algebras, hyper BE-algebras, hyper equality algebras, hyper hoops and etc. Moreover, they defined some new notions such as different kinds of hyper ideals, hyper filters and hyper congruence relations on these structures and studied some properties, the relation among them and the quotient structure. Now, in this paper, we review the definitions of all those hyper logical algebras and investigate relations among them.

State hyper BE-algebras

“…Also, it is similar that we can get the other part r • s ∩ [1] θ ̸ = ∅ for all s ∈ u • x, r ∈ u • y. This implies (u • x)∆ Ker(θ) (u • y).…”

“…Let θ be a hyper congruence relation on a hyper BE-algebra H. [1] θ is said to be the kernel of θ and is denoted by Ker(θ).…”

“…Theorem 2.9. [4] Let (H, •, 1) be a hyper BE-algebra and θ be a hyper congruence on H. Then (H/θ; •, [1] θ ) is a hyper BE-algebra, which is called as a quotient hyper BE-algebra with respect to θ.…”

“…Lemma 5.9. Let H be a transitive hyper BE-algebra and F be a hyper filter of H. Then [1] F is a hyper filter and Ker(∆…”

“…Let (H, ξ) be a state R-hyper BE-algebra and θ is a state compatible hyper congruence on (H, ξ). Then Ker(θ) = [1] θ is a state strong regular •-reflexive hyper filter of (H, ξ).…”