Biideals in BCK/BCI-Bialgebras

. The biideal structure in BCK/BCI-bialgebras is discussed. Relationships be-tween sub-bialgebras, biideals and IC-ideals (and/or CI-ideals) are considered. Conditionsfor a biideal to be a sub-bialgebra are provided, and conditions for a subset to be a biideal(resp. IC-ideal, CI-ideal) are given. 1. IntroductionA BCK/BCI-algebra is an important class of logical algebras introduced by K.Is´eki and was extensively investigated by several researchers. Bialgebraic struc-tures, for example, bisemigroups, bigroups, bigroupoids, biloops, birings, bisemir-ings, binear-rings, etc., are discussed in [4]. In [2], Jun et al. established thestructure of BCK/BCI-bialgebras, and investigated some properties. In this pa-per, we introduce the notion of biideals, IC-ideals and/or CI-ideals in BCK/BCI-bialgebras. We discuss relationships between biideals, IC-ideals (and/or CI-ideals)and sub-bialgebras, and give conditions for a biideal to be a sub-bialgebra. We alsoprovide conditions for a subset to be a biideal (resp. IC-ideal, CI-ideal).2. PreliminariesAn algebra (X;∗,0) of type (2,0) is called a BCI-algebra if it satises thefollowing conditions:(I) (∀x,y,z ∈ X) (((x∗y)∗(x∗z))∗(z ∗y) = 0),(II) (∀x,y ∈ X) ((x∗(x∗y))∗y = 0),(III) (∀x ∈ X) (x∗x = 0),(IV) (∀x,y ∈ X) (x∗y = 0,y ∗x = 0 ⇒ x = y).If a BCI-algebra X satises the following identity:(V) (∀x ∈ X) (0∗x = 0),