Normalizations of fuzzy BCC-ideals in BCC-algebras (Q2724022)

A groupoid \((X,\cdot)\) is called a BCC-algebra if there is a constant \(0\in X\) such that the following schemata: \(((x\cdot y)\cdot(z\cdot y))\cdot (x\cdot z)=0\), \(x\cdot x=0\), \(0\cdot x=0\), \(x\cdot 0=x\), and the rule: if \(x\cdot y=0\) and \(y\cdot x=0\), then \(x=y\), hold in \(X\). (If in addition \((x\cdot y)\cdot z=(x\cdot z)\cdot y\) holds in \(X\), then \(X\) is a BCK-algebra.) A fuzzy BCC-ideal of a BCC algebra \(X\) is a fuzzy set \(\mu :X\rightarrow [0,1]\) such that \(\mu(0)\geq\mu(x)\) for every \(x\in X\), and \(\mu(x\cdot z)\geq \min\{\mu((x\cdot y)\cdot z),\mu(y)\}\) for every \(x,y,z\in X\). The authors give some unary operations on the set of fuzzy BCC-ideals of \(X\). The notions of normal fuzzy BCC-ideal, maximal fuzzy BCC-ideal and completely normal fuzzy BCC-ideal are introduced. Among the other properties it is proven that every nonconstant normal fuzzy BCC-ideal which is maximal in the set of all normal fuzzy BCC-ideals ordered by inclusion takes only the values \(0\) and \(1\), and that every maximal fuzzy BCC-ideal is completely normal.