Characterization of lattice-valued multiset finite automata

Fuzzy Multiset Finite Automata: Some Algebraic and Lattice Theoretic Characterizations

“…The topological concepts already discussed in the case of (classical/fuzzy) automata (cf., [98][99][100]), the concepts of products and generalized products are well studied in the case of classical/fuzzy automata are remained to be explored in case of fuzzy multiset finite automata, we have been worked on these problems and ready to submit the related manuscripts. Other directions of future scope of study done in this paper are to study minimal realization of fuzzy multiset finite automata, where membership structure of fuzzy sets may be algebraic structures different from [0, 1] and distributive lattices keeping in the mind the fact that the nature of input sets (crisp set [19,20], fuzzy sets [101], multisets [47,51]) and structure of membership values ([0,1][20], poset, distributive lattice [102], residuated lattice [103,104], LSET [47]) of fuzzy automata play a very important role in characterization of various concepts in different versions of fuzzy automata, i.e., the properties of fuzzy automata which hold with one membership structure of fuzzy sets may not hold with other membership structures of fuzzy sets, e.g., categorical characterizations of concepts associated with fuzzy multiset finite automata studied in sections 5 and onwards of [47] do not simply holds if we change membership structure of fuzzy sets from LSET to any one of the structures [0, 1], arbitrary sets, posets, distributive lattice or complete residuated lattices because of role of functor U defined in proposition 10 of [47]. The relationship of categorical concepts with automata theory (cf., [62,[105][106][107][108][109][110]) and partial order sets [105]) are well known, such study may be carried out in case of FMFA and posets/lattice structures associated with FMFA introduced in this paper.…”