Algebraic and graph-theoretic properties of infinite n-posets
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 1, pp. 305-322.

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 n ω-ary product operations. Moreover, the ω-ary product operations give rise to n ω-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.

DOI : 10.1051/ita:2005018
Classification : 68Q45, 68R99
Mots clés : poset, $n$-poset, composition, free algebra, equational logic
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     title = {Algebraic and graph-theoretic properties of infinite $n$-posets},
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Ésik, Zoltán; Németh, Zoltán L. Algebraic and graph-theoretic properties of infinite $n$-posets. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 1, pp. 305-322. doi : 10.1051/ita:2005018. http://archive.numdam.org/articles/10.1051/ita:2005018/

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