Fundamental relation on hyper BCI-algebras
DOI:
https://doi.org/10.31489/2026m3/121-129Keywords:
algebraic hyperstructure, hypergroupoid, hyperoperation, BCK-algebra, hyper BCI-alebra, ideal, fundamental relation, BCI-algebra, hyper BCK-algebraAbstract
This paper investigates several structural aspects of hyper BCI-algebras and their associated equivalence relations. In particular, we study a notion related to families of hyper BCI-algebras, provide illustrative examples, and analyze the relation β defined on these algebraic systems. Special attention is devoted to its transitive closure, denoted by β∗, and its role in the general theory of hyper BCI-algebras. We prove that β∗ is the smallest equivalence relation for which the corresponding quotient structure is a classical BCI-algebra. Thus, β∗ can naturally be regarded as the fundamental relation associated with a hyper BCI-algebra. We also establish a number of conditions that are equivalent to the transitivity of the relation β, thereby providing further characterizations of the algebraic structures under consideration and extending existing results in the theory of hyper BCI-algebras. Several examples are included to illustrate the introduced concepts and the obtained results.
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