Geodesic
On the recognition of the finite definiteness of an automation monomial algebra
Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 2, pp. 503-516.

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In this paper an algorithm for recognition of finite definiteness of an automaton monomial algebra is proposed. It is shown that this problem for an arbitrary algebra reduces to the following problems: determination of the star height of a regular language and finite definiteness recognition for a certain class of automaton algebras. The solution of the former problem has already been described in the literature, the complete solution of the latter problem is presented in this paper. 
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D. A. Matsnev. On the recognition of the finite definiteness of an automation monomial algebra. Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 2, pp. 503-516. http://geodesic.mathdoc.fr/item/FPM_2002_8_2_a7/

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[2] A. Salomaa, Zhemchuzhiny teorii formalnykh yazykov, Mir, M., 1986 | MR

[3] E. V. Lukoyanova, Raspoznavaemye svoistva konechno opredelennykh assotsiativnykh algebr, Diss. $\dots$ kand. f.-m. nauk, Ulyanovsk, 1996

[4] K. Hashiguchi, “Algorithms for determining relative star height and star height”, Inform. and Comput., 78:2 (1988), 124–169 | DOI | MR | Zbl