Some results on ideals of BCK-algebras (Q2747286)

The author defines new subsets \([a; b^k]\) of a BCK-algebra \(X\) as NEWLINE\[NEWLINE[a; b^k]= \{x\in X\mid(x* a)* b^k= 0\},NEWLINE\]NEWLINE where \(x* y^k= (\cdots((x* y)* y)* y\cdots)* y\), and investigates some properties of these subsets.NEWLINENEWLINENEWLINEHe proves as main results the following:NEWLINENEWLINENEWLINE(1) Every ideal \(I\) is represented by the union of these sets, that is, \(I= \bigcup_{a,b\in I}[a; b^k]\).NEWLINENEWLINENEWLINE(2) The set \([a; b^k]\) is always a quasi-ideal, that is, it satisfies the following conditions: \(0\in [a; b^k]\) and if \(x\in [a; b^k]\) and \(y\in X\) then \(x* (x* y)\), \(y*(y* x)\in [a; b^k]\).NEWLINENEWLINENEWLINEThis means that each ideal can be represented as a union of quasi-ideals.