Hostname: page-component-7b7567c8c-5zcwv Total loading time: 0 Render date: 2026-11-15T08:59:30.760Z Has data issue: false hasContentIssue false
  • English
  • Français

Algebraic and graph-theoretic propertiesofinfinite 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
Get access

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 naturallyequipped with n binary product operations andnω-ary product operations.Moreover, the ω-ary product operationsgive rise to nω-power operations.We show that those Σ-labeled n-posets that can be generated fromthe singletons by the binary and ω-aryproduct operations form the free algebra on Σin a variety axiomatizable by an infinite collection of simpleequations. When n = 1, this variety coincides with the class ofω-semigroups of Perrin and Pin.Moreover, we show that those Σ-labeledn-posets that can be generated fromthe singletons by the binary product operations andthe ω-power operations form the free algebra on Σin a related variety that generalizes Wilke's algebras.We also give graph-theoretic characterizationsof those n-posets contained in the above free algebras. Our resultsserve as a preliminary study to a development of a theory ofhigher dimensional automata and languages on infinitaryassociative structures.

Keywords

Posetn-posetcompositionfree algebraequational logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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