On minimality in pseudo-$BCI$-algebras
Commun. Korean Math. Soc. 2012 Vol. 27, No. 1, 7-13
Printed March 1, 2012
Young Hee Kim and Keum Sook So
Chungbuk National University, Hallym University
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
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