\((\alpha, \beta)\)-fuzzy Lie algebras over an \((\alpha, \beta)\)-fuzzy field (Q991811)

The authors consider \((\alpha,\beta)\)-fuzzy Lie algebras over \((\alpha,\beta)\)-fuzzy fields. The notion of an \((\alpha,\beta)\)-fuzzy algebraic structure arose from the concept of quasi-coincidence of a fuzzy point with a fuzzy subset. A fuzzy point \(x_t\) is said to belong to a fuzzy subset \(A\), written \(x_t\in A\), if \(A(x)\geq t\). A fuzzy point \(x_t\) is said to be quasi-coincident with a fuzzy subset \(A\), written \(x_t qA\), if \(A(x)+ t> 1\). Let \(\alpha\) and \(\beta\) denote any one of \(\varepsilon\), \(q\), \(\varepsilon\vee q\), where \(\varepsilon\vee q\) means \(\varepsilon\) or \(q\). For a fuzzy subset \(A\) of a set \(S\), let \(A_t= \{x\in S\mid A(x)\geq t\}\). Let \(X\) be a field and let \(F: X\to[0,1]\) be a fuzzy subset of \(X\). Then \(F\) is called an \((\alpha,\beta)\)-fuzzy subfield of \(X\) if the following conditions hold for all \(t,s\in[0,1]\): (i) For all \(x,y\in X\), \(x_t\alpha F\), \(y_s\alpha F\) implies \((x- y)_{t\wedge s}\beta F\), where \(\wedge\) denotes minimum; (ii) For all \(x,y\neq 0\) in \(X\), \(x_t\alpha JF\), \(y_s\alpha F\) implies \((xy^{-1})_{t\wedge s}\beta F\). Let \(L\) be a Lie algebra over a field \(X\) and let \(F\) be an \((\alpha,\beta)\)-fuzzy subfield of \(X\). A fuzzy subset \(A: L\to[0,1]\) of \(L\) is called an \((\alpha,\beta)\)-fuzzy Lie algebra of \(L\) over \(F\) if it satisfies the following conditions for all \(a,b\in L\), \(x\in X\), \(t,s,r\in[0, 1]\): (i) \(a_t\alpha A\), \(b_s\alpha A\) implies \((a-b)_{t\wedge s}\beta A\); (ii) \(a_t\alpha A\), \(x_r\alpha F\) implies \((xa)_{t\wedge r}\beta A\); (iii) \(a_t\alpha A\), \(b_s\alpha A\) implies \(([a, b])_{t\wedge s}\beta A\), where \([a,b]\) denotes the Lie bracket. The authors prove that if \(A\) is an \((\varepsilon,\varepsilon\vee q)\)-fuzzy Lie algebra of a Lie algebra \(L\) over an \((\varepsilon, \varepsilon\vee q)\)-fuzzy field \(F\) of a field \(X\), then for \(t\in(0,5]\), \(A_t\) is a Lie subalgebra over \(F_t\) when \(F_t\) contains at least two elements. The authors also show that homomorphic images of \((\varepsilon,\varepsilon\vee q)\)-fuzzy Lie algebras arc \((\varepsilon, \varepsilon\vee q)\)-fuzzy Lie algebras and similarly for pre-images of homomorphisms.