Generalized fuzzy filters in non-commutative residuated lattices.
The paper defines and studies some properties of a so-called notion of \((\alpha,\beta)_T\)-\textit{fuzzy filter} over bounded integral residuated lattices, not necessarily commutative. The notion of \((\alpha,\beta)_T\)-fuzzy filter is defined as follows. \(\alpha\) and \(\beta\) are both members of a four element set, providing the typology of the fuzzy filter. In the paper, these four typologies for \(\alpha\) and for \(\beta\) are denoted: \(\in\), \(q\), \(\in\wedge q\) and \(\in\vee q\), where \(q\), for the author, means \textit{quasi-coincidence}. Let \(L\) be a bounded integral residuated lattice. Given \(\alpha\) of one of the four typologies above, a fuzzy subset \(\mu\colon L\to [0,1]\) of \(L\), and \(r\in (0,1]\), the author defines the notion of \(x_r\alpha\mu\) according to the typology of \(\alpha\), as follows. type 1: \(x_r\alpha\mu\) iff \(\mu(x)\geq r\); type 2: \(x_r\alpha\mu\) iff \(\mu(x)+r>1\); type 3: \(x_r\alpha\mu\) iff \(\mu(x)\geq r\) and \(\mu(x)+r>1\); type 4: \(x_r\alpha\mu\) iff \(\mu(x)\geq r\) or \(\mu(x)+r>1\). Fix \(\alpha\) and \(\beta\). Let \(L\) be a bounded integral residuated lattice and \(T\) be a t-norm. Let \(\mu\colon L\to [0,1]\) be a fuzzy subset of \(L\). Then \(\mu\) is an \((\alpha,\beta)_T\)-fuzzy filter if for all \(r,s\in (0,1]\) and \(x,y,z\in L\) the following hold: (1) \(x_r\alpha\mu\) and \(y_s\alpha\mu\) imply \((x*y)_{T(r,s)}\beta\mu\) and \((y*x)_{T(r,s)}\beta\mu\); (2) \(x_r\alpha\mu\) and \(x\leq y\) imply \(y_r\beta\mu\). The author studies the notion arising by some combinations of \(\alpha\) and \(\beta\) exhibiting properties and examples.
scientific article; zbMATH DE number 6366059
scientific article