On characterizations of regular ordered semigroups in terms of \(\mathcal Q\)-fuzzy subsets
According to this paper, if \(S\) is an ordered semigroup and \(Q\) a nonempty set, then any mapping \(f\) from \(S\times Q\) into the closed interval \([0,1]\) of real numbers is called a \(Q\)-fuzzy subset of \(S\). A \(Q\)-fuzzy subset of \(S\) is called a \(Q\)-fuzzy left (right) ideal of \(S\) if (1) \(x\leq y\) implies \(f(x,q)\geq f(y,q)\) for all \(q\in Q\) and (2) if \(f(xy,q)\geq f(y,q)\) (resp. \(f(xy,q)\geq f(x,q)\)) for all \(x,y\in S\) and all \(q\in Q\). A \(Q\)-fuzzy subset of \(S\) is called a \(Q\)-fuzzy generalized bi-ideal of \(S\) if (1) \(x\leq y\) implies \(f(x,q)\geq f(y,q)\) for all \(q\in Q\) and (2) if \(f(xyz,q)\geq \min\{f(x,q),f(z,q)\}\) for all \(x,y,z\in S\) and all \(q\in Q\). In this paper the author transfers the results given by \textit{X. Y. Xie} and \textit{J. Tang} [Iran. J. Fuzzy Syst. 7, No. 2, 121--140 (2010 Zbl 1223.06013)] to the case of \(Q\)-fuzzy subsets, exactly in the way they are given by Xie and Tang.NEWLINENEWLINE But the results of the paper by Lekkoksung are actually the results by Xie and Tang in [loc. cit.]. The study of \(Q\)-fuzzy subsets does not add any information on fuzzy sets (subsets). In fact this \(q\) plays no role in the study of fuzzy sets. I mean, if one gets any result on fuzzy subset and adds that \(q\), then one gets the corresponding result on \(Q\)-fuzzy subsets. And conversely, if one gets any result on \(Q\)-fuzzy subsets and removes that \(q\), then one gets its corresponding one on fuzzy subsets.
scientific article; zbMATH DE number 6233542