On pseudo-fuzzy linear mappings

In this paper we study a more general class of linear maps on fuzzy subsets of vector spaces than those on fuzzy subspaces. We obtain a version of the fundamental theorem of homomorphisms which applies to these pseudo-fuzzy linear maps. In the case of fuzzy subspaces, the results become those for fuzzy linear maps. We introduce the notion of magnification of pseudo-fuzzy linear mappings. It is shown that this notion behaves well with respect to the fundamental theorem and with respect to the composition of mapping. We note that the class of distribution functions of random variables naturally produces pseudo-fuzzy linear mappings, which are not fuzzy linear mappings. We observe that pseudo-fuzzy linear mappings are examples of gain relations. We consider general properties of these relations as well as many other examples. From these results additional properties of pseudo-fuzzy linear mappings are obtained.