Fuzzy vector spaces under triangular norms
Let E be a vector space over a field K and F a fuzzy set defined on E. According to \textit{A. K. Katsaras} and \textit{D. B. Liu} [J. Math. Anal. Appl. 58, 135-146 (1977; Zbl 0358.46011)] F is a fuzzy subspace of E if \(F(x+y)\leq \min (F(x),F(y))\) and F(kx)\(\geq F(x)\) for all x,y\(\in E\) and \(k\in K\). In this paper the author uses a t-norm instead of minimum. Many theorems obtained are generalizations of the corresonding results by Katsaras and Liu. However, this generalization of a fuzzy subspace was introduced earlier by \textit{Yu Yandong} [Fuzzy Math. 4, No.2, 29-40 (1984; Zbl 0565.46008) and ibid. No.3, 59-62 (1984; Zbl 0565.46009)]. These papers also contain several elementary theorems, which are proved here.
scientific article; zbMATH DE number 4026315