The lattice of \(F^A_R\)-modules (Q2702838)

The authors elaborate on the notion of a fuzzy module introduced by \textit{C. V. Negoita} and \textit{D. A. Ralescu} in their pioneering 1975 book [on the Applications of Fuzzy Sets to System Analysis (Birkhäuser-Verlag, Basel, 1975; Zbl 0326.94002)]. The connection between a fuzzy module and a fuzzy ring has been established by J. Zhao in introducing the so-called \(F^A_R(M)\)-modules, i.e. fuzzy submodule of a left \(R\)-module \(M\) over a fuzzy subgring \(A\) of a ring \(R\).NEWLINENEWLINENEWLINEHere, the authors investigate these \(F^A_R\)-modules from a lattice theoretic point of view. In particular, the following properties have been proved: -- \(F^A_R(M)\) forms a complete lattice w.r.t. Zadeh's sharp inclusion relation; -- the sum or sup-min composition of two \(F^A_R\)-modules of \(M\) also constitutes an \(F^A_R\)-module; -- \(F^A_R(M)\) constitutes a modular lattice; -- the direct and inverse image of an \(F^A_R\)-module under a homomorphism constitute again \(F^A_R\)-modules. -- Finally it is stated, however without proof, that the lattice \(F^A_R(M)\) is not distributive.