An Effective Lower Bound for Group Complexity of Finite Semigroups and Automata

TL;DR

This paper introduces an effective lower bound for the group complexity of finite semigroups and automata, addressing a longstanding computational problem in algebraic automata theory.

Contribution

It provides a novel, computable lower bound for the group complexity, advancing the understanding of algebraic structure in finite automata and semigroups.

Findings

Established an effective lower bound for group complexity

Improved understanding of algebraic decomposition in automata

Contributed to the theoretical foundation of automata complexity analysis

Abstract

The question of computing the group complexity of finite semigroups and automata was first posed in K. Krohn and J. Rhodes, \textit{Complexity of finite semigroups}, Annals of Mathematics (2) \textbf{88} (1968), 128--160, motivated by the Prime Decomposition Theorem of K. Krohn and J. Rhodes, \textit{Algebraic theory of machines, {I}: {P}rime decomposition theorem for finite semigroups and machines}, Transactions of the American Mathematical Society \textbf{116} (1965), 450--464. Here we provide an effective lower bound for group complexity.