×

Partially ordered sets and the independence property. (English) Zbl 0699.03020

Summary: No theory of a parially ordered set of finite width has the independence property, generalizing Poizat’s corresponding result for linearly ordered sets. In fact, a question of Poizat concerning linearly ordered sets is answered by showing, moreover, that no theory of a partially ordered set of finite width has the multi-order property. It then follows that a distributive lattice is not finite-dimensional iff its theory has the independence property iff its theory has the multi-order property.

MSC:

03C65 Models of other mathematical theories
06A06 Partial orders, general
06D99 Distributive lattices
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Théories d’arbres 47 pp 841– (1982)
[2] The stability function of a theory 43 pp 481– (1978)
[3] DOI: 10.1090/S0002-9947-1984-0742419-0
[4] DOI: 10.2307/1969503 · Zbl 0038.02003
[5] DOI: 10.1016/0003-4843(71)90015-5 · Zbl 0281.02052
[6] Théories instables 46 pp 513– (1981)
[7] Proceedings of the American Mathematical Society 87 pp 707– (1983)
[8] Decidability and finite axiomatizability of theories of 0-categorical partially ordered sets 46 pp 101– (1981)
[9] Decidability and 0-categoricity of theories of partially ordered sets 45 pp 585– (1980) · Zbl 0441.03007
[10] Transactions of the American Mathematical Society 141 pp 1– (1969)
[11] Orders: description and roles (proceedings, L’Arbresle, 1982 pp 269– (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.