The algebra of row monomial matrices

TL;DR

This paper explores a novel algebraic structure based on row monomial matrices with a special summation operation, applying it to automata theory and providing insights related to the Cerny conjecture.

Contribution

It introduces a new algebraic framework for row monomial matrices with a unique summation operation, linking it to automata theory and the Cerny conjecture.

Findings

Defined a summation operation for row monomial matrices

Established closure properties under the new operation

Applied the algebra to automata synchronization and the Cerny conjecture

Abstract

We consider an algebra with non-standard operations on the class of row monomial matrices (having one unit and rest of zeros in every row). The class of row monomial matrices is closed under multiplication, but not closed under ordinary matrix addition. The paper considers a kind of summation operation on row monomial matrices and the necessary conditions to be closed under the operation in this class. The most significant difference between the algebra of row monomial matrices and linear algebra is the summation operation, with respect to which the class of row monomial matrices is closed. The operation of summation in the algebra can be considered also as an algebra of subsets of any set. The class of subsets of given set is closed under considered operation of summation. The deterministic finite automaton (DFA) can be presented by a complete underlying graph of the automaton with edges labelled by letters of an alphabet. Row monomial matrices can be induced by words in the alphabet of labels on edges of underlying graph of DFA and present a mapping of the set of states. A word $s$ of letters on edges of underlying graph $\Gamma$ of deterministic finite automaton (DFA) is called synchronizing if $s$ sends all states of the automaton to a unique state. J. \v{C}erny discovered in 1964 a sequence of $n$-state complete DFA possessing a minimal synchronizing word of length $(n-1)^2$. The hypothesis, mostly known today as \v{C}erny conjecture, claims that $(n-1)^2$ is a precise upper bound on the length of such a word over alphabet $\Sigma$ of letters on edges of $\Gamma$ for every complete $n$-state DFA. The hypothesis was formulated in 1966 by Starke. The following proof of the conjecture uses the algebra of row monomial matrices.