On Fuzzy Proper Exact Sequences and Fuzzy Projective Semimodules Over Semirings

Abstract

As an analogue here we extend and give new horizon to semimodule theory by introducing fuzzy exact and proper exact sequences of fuzzy semi modules for generalizing well known theorems and results of semimodule theory to their fuzzy environment. We also elucidate completely the characterization of fuzzy projective semi modules via Hom functor and show that semimodule µP is fuzzy projective if and only if Hom(µP ,–) preservers the exactness of the sequence µM′ α¯−→νM β¯ −→ηM′′ with β¯ being K-regular. Some results of commutative diagram of R-semimodules having exact rows specifically the “5-lemma” to name one, were easily transferable with the novel proofs in their fuzzy context. Also, towards the end apart from the other equivalent conditions on homomorphism of fuzzy semimodules it is necessary to see that in semimodule theory every fuzzy free is fuzzy projective however the converse is true only with a specific condition.