Galois theory for fuzzy fields
Let \(F/K\) be a field extension and let \(G(F/K)\) denote the Galois group of \(F/K\). In this paper the author develops a generalized form of the Galois connection between the intermediate fields of a field extension and the subgroups of \(G(F/K)\). In particular, the author extends the Galois connection to include fuzzy intermediate fields of \(F/K\) and fuzzy subgroups of \(G(F/K)\). Let \(L\) denote the set of all fuzzy intermediate fields of \(F/K\) and \(H\) the set of all fuzzy subgroups of \(G(F/K)\). A Galois connection between \(L\) and \(H\) is defined to be a pair of functions \((\Phi,\Psi)\) such that \(\Phi:L\to H\), \(\Psi:H\to L\) and \(\Phi\Psi(\mu) \geq\mu\) \(\forall\mu\in H\), \(\Psi \Phi (\eta)\geq \eta\) \(\forall\eta \in L\). Let \(x\in F\) and \(\sigma\in G(F/K) \). Define \(A_x=\{\sigma\in G(F/K)\mid \sigma(x)\neq x\}\) and \(A_\sigma= \{x\in F \mid\sigma (x)\neq x\}\). The author shows that if \((\Phi,\Psi)\) is defined by \(\Psi(\mu) (x)=1-\sup\{\mu (\sigma)\mid \sigma\in A_x\}\) \(\forall x\in F\) and \(\Phi(\eta) (\sigma)=1 -\sup\{\eta (x)\mid x\in A_\sigma\}\) \(\forall\sigma\in G(F/K)\), then \((\Phi,\Psi)\) is a Galois connection between \(L\) and \(H\). The author also shows that if \(F/K\) is a Galois extension with \([F:K] <\infty\), then there is a one-to-one correspondence between the fuzzy intermediate fields of \(F/K\) and the fuzzy subgroups of \(G(F/K)\) given by \(\eta\to\Phi(\eta)\). The inverse of this function is \(\mu\to\Psi (\mu)\). The interested reader should note that a fuzzy Galois theory has been developed by the reviewer [J. Fuzzy Math. 1, No. 3, 659-671 (1993; Zbl 0787.12003) and Infinite fuzzy Galois theory, Adv. Fuzzy Theory Technology, Vol. II, 99-109 (1994; Zbl 0787.12003)]. See also, \textit{J. N. Mordeson} and \textit{D. S. Malik}, Fuzzy Commutative Algebra, World Scientific (1998).
scientific article; zbMATH DE number 1563546