Exact sequences of fuzzy linear maps
The paper deals with ``fuzzy'' versions of standard homological facts from the theory of modules (vector spaces); namely: the first isomorphism theorem for short exact sequences for fuzzy linear maps (f.l.m.) over fuzzy vector spaces the third isomorphism theorem and the 5-lemma for quasi-monic epic f.l.m, the stability property for an infinite complex of f.l.m., the property of split exactness of a short sequence and so on. Necessary concepts for these versions are introduced by the author in terms of the ``underlying'' ordinary linear map and the usual map \(\chi\) :V\(\to [0,1]\) for a fuzzy vector space \(\chi_ V.\) The key concept is the following: f.l.m. \(\tilde f:\)\(\chi_ V\to \eta_ W\) is said to be quasi-monic if \(\chi_{\ker}\tilde f=\chi_{V^*}\), where \(V^*=\{v\in V| \chi (v)=1\}\) and ker \(\tilde f:=\{v\in V| \eta (f(v))=1\}.\) Just the concept admits the technique of diagram search. Also, the author intends to set up the functor theory of fuzzy vector spaces (modules) in the near future.
fuzzy versions of standard homological facts from the theory of modules
first isomorphism theorem for short exact sequences for fuzzy linear maps
scientific article; zbMATH DE number 4080187