Fuzzifications of ideals in BCC-algebras (Q2748021)

The authors extend the concepts of BCC-ideal and \(g\)-ideal to fuzzy BCC-ideals and fuzzy \(g\)-ideals, respectively, and show that for a fuzzy set \(\mu\) in a BCC-algebra, \(\mu\) is a fuzzy \(g\)-ideal if and only if it is a fuzzy BCC-ideal. In particular, for a fuzzy set \(\mu\) in a BCK-algebra, \(\mu\) is a fuzzy \(g\)-ideal if and only if it is a fuzzy BCK-ideal. Thus three kinds of fuzzy ideals (fuzzy \(g\)-ideals, fuzzy BCC-ideals, and fuzzy BCK-ideals) coincide in the case of BCK-algebras.NEWLINENEWLINENEWLINEThe authors also consider a direct product \(\mu\times\nu\) of fuzzy sets \(\mu\), \(\nu\) and prove that:NEWLINENEWLINENEWLINE1. \(\mu\), \(\nu\) fuzzy \(g\)-ideals \(\Rightarrow \mu\times \nu\) fuzzy \(g\)-ideal (Theorem 5.9).NEWLINENEWLINENEWLINE2. \(\mu\times \nu\) fuzzy \(g\)-ideal \(\Rightarrow\) either \(\mu\) or \(\nu\) is a fuzzy \(g\)-ideal (Theorem 5.10).NEWLINENEWLINENEWLINE3. \(\nu\times\nu\) fuzzy \(g\)-ideal \(\Leftrightarrow\nu\) fuzzy \(g\)-ideal (Theorem 5.11).NEWLINENEWLINENEWLINEThe last theorem comes from the fact that \(\nu\times\nu(x, x)= \nu(x)\) or the two previous theorems 5.9 and 5.10 immediately.