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Algebraic and graph-theoretic properties ofinfinite n-posets

Published online by Cambridge University Press:  15 March 2005

Zoltán Ésik  and
Zoltán L. Németh
Zoltán Ésik
Affiliation:
Department of Computer Science, University of Szeged, P.O.B. 652, 6701 Szeged, Hungary; zlnemeth@inf.u-szeged.hu
Zoltán L. Németh
Affiliation:
Department of Computer Science, University of Szeged, P.O.B. 652, 6701 Szeged, Hungary; zlnemeth@inf.u-szeged.hu
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Abstract

A Σ-labeled n-poset is an (at most) countable set, labeled in the set Σ, equipped with n partial orders. The collection of all Σ-labeled n-posets is naturally equipped with n binary product operations and nω-ary product operations. Moreover, the ω-ary product operations give rise to nω-power operations. We show that those Σ-labeled n-posets that can be generated from the singletons by the binary and ω-ary product operations form the free algebra on Σ in a variety axiomatizable by an infinite collection of simple equations. When n = 1, this variety coincides with the class of ω-semigroups of Perrin and Pin. Moreover, we show that those Σ-labeled n-posets that can be generated from the singletons by the binary product operations and the ω-power operations form the free algebra on Σ in a related variety that generalizes Wilke's algebras. We also give graph-theoretic characterizations of those n-posets contained in the above free algebras. Our results serve as a preliminary study to a development of a theory of higher dimensional automata and languages on infinitary associative structures.

Keywords

Posetn-posetcompositionfree algebraequational logic
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 39 , Issue 1: Imre Simon, the tropical computer scientist , January 2005 , pp. 305 - 322
Copyright
© EDP Sciences, 2005

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References

Bloom, S.L. and Ésik, Z., Shuffle binoids. Theoret. Inform. Appl. 32 (1998) 175–198. CrossRef
Z. Ésik, Free algebras for generalized automata and language theory. RIMS Kokyuroku 1166, Kyoto University, Kyoto (2000) 52–58.
Z. Ésik and Z.L. Németh, Automata on series-parallel biposets, in Proc. DLT'01. Lect. Notes Comput. Sci. 2295 (2002) 217–227.
Z. Ésik and S. Okawa, Series and parallel operations on pomsets, in Proc. FST & TCS'99. Lect. Notes Comput. Sci. 1738 (1999) 316–328.
Gischer, J.L., The equational theory of pomsets. Theoret. Comput. Sci. 61 (1988) 199–224. CrossRef
Grabowski, J., On partial languages. Fund. Inform. 4 (1981) 427–498.
K. Hashiguchi, S. Ichihara and S. Jimbo, Formal languages over free binoids. J. Autom. Lang. Comb. 5 (2000) 219–234.
Hashiguchi, K., Wada, Y. and Jimbo, S., Regular binoid expressions and regular binoid languages. Theoret. Comput. Sci. 304 (2003) 291–313. CrossRef
D. Kuske, Infinite series-parallel posets: logic and languages, in Proc. ICALP 2000. Lect. Notes Comput. Sci. 1853 (2001) 648–662.
D. Kuske, A model theoretic proof of Büchi-type theorems and first-order logic for N-free pomsets, in Proc. STACS'01. Lect. Notes Comput. Sci. 2010 (2001) 443–454.
Kuske, D., Towards a language theory for infinite N-free pomsets. Theoret. Comput. Sci. 299 (2003) 347–386. CrossRef
K. Lodaya and P. Weil, Kleene iteration for parallelism, in Proc. FST & TCS'98. Lect. Notes Comput. Sci. 1530 (1998) 355–366.
Lodaya, K. and Weil, P., Series-parallel languages and the bounded-width property. Theoret. Comput. Sci. 237 (2000) 347–380. CrossRef
Lodaya, K. and Weil, P., Rationality in algebras with series operation. Inform. Comput. 171 (2001) 269-293. CrossRef
D. Perrin and J.-E. Pin, Semigroups and automata on infinite words, in Semigroups, Formal Languages and Groups (York, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 466 (1995) 49–72.
D. Perrin and J.-E. Pin, Infinite Words. Pure and Applied Mathematics 141, Academic Press (2003).
Valdes, J., Tarjan, R.E. and Lawler, E.L., The recognition of series-parallel digraphs. SIAM J. Comput. 11 (1982) 298–313. CrossRef
Th. Wilke, An algebraic theory for regular languages of finite and infinite words. Internat. J. Algebra Comput. 3 (1993) 447–489. CrossRef