A Unique Operator-Topological Decomposition Framework for Lℬ-Valued General Fuzzy Automata

Abstract

[Formula: see text]-valued general fuzzy automata provide a rich algebraic–topological framework for modeling systems with uncertainty and graded dynamics. This paper investigates the structural decomposition of [Formula: see text]-valued GFA through the lens of three fundamental operators: the successor operator [Formula: see text], the predecessor operator [Formula: see text], and the core operator [Formula: see text]. These operators induce dual topologies on the state space and allow us to establish a canonical decomposition theorem. We prove that every [Formula: see text]-valued GFA can be uniquely decomposed into the union of a regular-closed decomposable subautomaton and a predecessor (source-splitting) subautomaton. This result generalizes classical decomposition theorems for crisp automata such as the Krohn–Rhodes theorem to the [Formula: see text]-valued setting. The proposed framework bridges algebra and topology and opens new perspectives for future applications in soft computing and intelligent system modeling.