Abstract
In this note, we focus again on the basics of triangular product of automata construction and to introduce the notion of linear automata complexity. It contains three main results. (1) For any two pure automata we consider the category of their cascade connections. It possesses the universal terminal object. This object is the wreath product of the automata. Hence, every cascade connection admits a natural embedding into the wreath product of automata. (2) A similar theory is developed for linear automata, where we also consider the category of cascade connections. It also has the terminal object. This object is the triangular product of linear automata. (3) Triangular products have various applications. This construction is used in linear automata decomposition theory and in the definition of complexity of a linear automaton. We consider a special linear complexity and give the rule for its calculation.
Similar content being viewed by others
References
M. A. Arbib, ed., Algebraic Theory of Machines, Languages, and Semigroups, Academic Press, New York (1968).
S. Eilenberg, Automata, Languages and Machines, Academic Press, London (1976).
F. Gecseg, Products of Automata, EATCS Monogr. on TCS, Springer (1986).
F. Gecseg and I. Peak, Algebraic Theory of Automata, Akademiai Kiado (1972).
M. Holcombe, Algebraic Automata Theory, Cambridge Univ. Press, Cambridge (1982).
M. Ito, Algebraic Theory of Automata and Languages, World Scientific (2004).
M. Kambites, “On the Krohn–Rhodes complexity of semigroups of upper triangular matrices,” Int. J. Algebra Comput., 17, No. 1, 187–201 (2007).
M. Kambites and B. Steinberg, “Wreath product decompositions for triangular matrix semigroups,” in: Proc. Semigroups and Languages, Lisbon (2005), pp. 129–144.
K. Krohn and J. Rhodes, “Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines,” Trans. Am. Math. Soc., 116, 450–464 (1965).
K. Krohn and J. Rhodes, “Complexity of finite semigroups,” Ann. Math., 2, No. 88, 128–160 (1968).
K. Krohn, J. Rhodes, and B. Tilson, “Lectures on finite semigroups,” in: Algebraic Theory of Machines, Languages and Semigroups, Academic Press, New York (1968).
B. Plotkin, L. Greenglaz, and A. Gvaramija, Algebraic Structures in Automata and Databases Theory, World Scientific, Singapore (1992).
B. Plotkin and T. Plotkin, “An algebraic approach to knowledge bases informational equivalence,” Acta Appl. Math., 89, 109–134 (2005).
B. I. Plotkin and S. M. Vovsi, Varieties of Representations of Groups, Zinatne (1983).
J. Rhodes and B. Steinberg, The q-Theory of Finite Semigroups, to appear; http://mathstat.math.carleton.ca/~bsteinbg/qtheor.html.
S. M. Vovsi, Triangular Products of Group Representations and Their Applications, Progress Math., Vol. 17, Birkhäuser (1981).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 7, pp. 175–186, 2011/12.
Rights and permissions
About this article
Cite this article
Plotkin, B., Plotkin, T. Cascade Connections and Triangular Products of Linear Automata. J Math Sci 197, 565–572 (2014). https://doi.org/10.1007/s10958-014-1735-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-1735-0