On minimality in pseudo-$BCI$-algebras

Young Hee Kim; Keum Sook So

doi:10.4134/CKMS.2012.27.1.7

Communications of the
Korean Mathematical Society CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2012; 27(1): 7-13

Printed March 1, 2012

https://doi.org/10.4134/CKMS.2012.27.1.7

On minimality in pseudo-$BCI$-algebras

Young Hee Kim and Keum Sook So

Chungbuk National University, Hallym University

Abstract

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Abstract

In this paper we consider pseudo-$BCK/BCI$-algebras. In particular, we consider properties of minimal elements ($x\leq a$ implies $x=a$) in terms of the binary relation $\leq$ which is reflexive and antisymmetric along with several more complicated conditions. Some of the properties of minimal elements obtained bear resemblance to properties of $B$-algebras in case the algebraic operations $*$ and $\circ$ are identical, including the property $0\circ(0*a)= a$. The condition $0*(0\circ x) = 0\circ(0*x)= x$ for all $x\in X$ defines the class of $p$-semisimple pseudo-$BCK/BCI$-algebras ($0\leq x$ implies $x=0$) as an interesting subclass whose further properties are also investigated below.

Keywords: (pseudo-)$BCK/BCI$-algebra, minimal, $p$-semisimple

MSC numbers: 06F35