Recognition of Certain Properties of Automaton Algebras

The paper considers a new algebraic object, the completely automaton binomial algebras, which generalize certain existing classes of algebras. The author presents a classification of semigroup algebras taking into account completely automaton algebras and gives the corresponding examples. A number of standard algorithmic problems are solved for completely automaton binomial algebras: the recognition of a strict and non-strict polynomial property, the recognition of the right and/or left finite processing, and the construction of the determining regular language for an algebra with finite processing and for monomial subalgebras of a free associative algebra and certain completely automaton algebras. for an automaton monomial algebra, the author constructs the left syzygy module of a finite system of elements and the Gröbner basis of a finitely generated left ideal; also, some algorithmic problems are solved.