Abstract
We observe that, given a poset \({\left( {E,{\user1{\mathcal{R}}}} \right)}\) and a finite covering \({\user1{\mathcal{R}}} = {\user1{\mathcal{R}}}_{1} \cup \cdots \cup {\user1{\mathcal{R}}}_{n} \) of its ordering, the height of the poset does not exceed the natural product of the heights of the corresponding sub-relations:
Conversely for every finite sequence \((\xi_1,\cdots,\xi_n)\) of ordinals, every poset \({\left( {E,{\user1{\mathcal{R}}}} \right)}\) of height at most \(\xi_1\otimes\cdots\otimes\xi_n\) admits a partition \({\left( {{\user1{\mathcal{R}}}_{1} , \cdots ,{\user1{\mathcal{R}}}_{n} } \right)}\) of its ordering \({\user1{\mathcal{R}}}\) such that each \({\left( {E,{\user1{\mathcal{R}}}_{k} } \right)}\) has height at most \(\xi_k\). In particular for every finite sequence \((\xi_1,\cdots,\xi_n)\) of ordinals, the ordinal
is the least \(\xi\) for which the following partition relation holds
meaning: for every poset \({\left( {A,{\user1{\mathcal{R}}}} \right)}\) of height at least \(\xi\) and every finite covering \({\left( {{\user1{\mathcal{R}}}_{1} , \cdots ,{\user1{\mathcal{R}}}_{n} } \right)}\) of its ordering \({\user1{\mathcal{R}}}\), there is a \(k\) for which the relation \({\left( {A,{\user1{\mathcal{R}}}_{k} } \right)}\) has height at least \(\xi_k\). The proof will rely on analogue properties of vertex coverings w.r.t. the natural sum.
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References
Abraham, U.: A note on Dilworth’s theorem in the infinite case. Order 4, 117–125 (1987)
Abraham, U., Bonnet, R.: Hausdorff’s theorem for posets that satisfy the finite anti-chain property. Fundam. Math. 159(1), 51–69 (1999)
Carruth, P.W.: Arithmetic of ordinals with applications to the theory of ordered Abelian groups. Bull. Am. Math. Soc. 48, 262–271 (1942)
Delhommé, C.: Automaticity of ordinals and of homogeneous graphs (Automaticité des ordinaux et des graphes homogènes). C. R. Math. Acad. Sci. Paris 339(1), 5–10 (2004)
Delhommé, C.: Decomposition of tree-automatic structures. (2003) (Manuscript)
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Delhommé, C. Height of a Superposition. Order 23, 221–233 (2006). https://doi.org/10.1007/s11083-006-9044-y
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DOI: https://doi.org/10.1007/s11083-006-9044-y