Algebraic and graph-theoretic properties of infinite n-posets

Zoltán Ésik; Zoltán L. Németh

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 39, Issue: 1, page 305-322
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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Ésik, Zoltán, and Németh, Zoltán L.. "Algebraic and graph-theoretic properties of infinite n-posets." RAIRO - Theoretical Informatics and Applications 39.1 (2010): 305-322. <http://eudml.org/doc/92763>.

@article{Ésik2010,
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. },
author = {Ésik, Zoltán, Németh, Zoltán L.},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Poset; n-poset; composition; free algebra; equational logic},
language = {eng},
month = {3},
number = {1},
pages = {305-322},
publisher = {EDP Sciences},
title = {Algebraic and graph-theoretic properties of infinite n-posets},
url = {http://eudml.org/doc/92763},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Ésik, Zoltán
AU - Németh, Zoltán L.
TI - Algebraic and graph-theoretic properties of infinite n-posets
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 1
SP - 305
EP - 322
AB - 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.
LA - eng
KW - Poset; n-poset; composition; free algebra; equational logic
UR - http://eudml.org/doc/92763
ER -

References

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