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2008

The length of chains in algebraic lattices

Chakir, Ilham, Pouzet, Maurice

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We study how the existence in an algebraic lattice $L$ of a chain of a given type is reflected in the join-semilattice $K(L)$ of its compact elements.

We show that for every chain $\alpha$ of size $\kappa$, there is a set $\B$ of at most $2^{\kappa}$ join-semilattices, each one having a least element such that an algebraic lattice $L$ contains no chain of order type $I(\alpha)$ if and only if the join-semilattice $K(L)$ of its compact elements contains no join-subsemilattice isomorphic to a member of $\B$.
We show that among the join-subsemilattices of $[\omega]^{<\omega}$ belonging to $\B$, one is embeddable in all the others.
We conjecture that if $\alpha$ is countable, there is a finite set $\B$.

The length of chains in algebraic lattices · Around

Built on

G. Higman, Ordering by divisibility in abstract algebras, Proc. London. Math. Soc. 2 (3), (1952), 326-336
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R Bonnet, M Pouzet, Extensions et stratifications d’ensembles dispersés, Comptes Rendus Acad. Sc.Paris, 268, Série A, (1969),1512-1515
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M. Pouzet and N. Zaguia, Ordered sets with no chains of ideals of a given type, Order, 1
1984
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I.Chakir, Chaînes d’idéaux et dimension algébrique des treillis distributifs, Thèse de doctorat, Université Claude-Bernard(Lyon1) 18 décembre 1992, n¡ 1052
1992
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Similar

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I.Chakir, M.Pouzet, The length of chains in distributive lattices, Notices of the A.M.S., 92 T-06-118, 502-503
Cited in the paper.
I. Chakir, M. Pouzet, A characterization of well-founded algebraic lattices, submitted to Order, under revision
Cited in the paper.

Then

I.Chakir, M.Pouzet, Infinite independent sets in distributive lattices, Algebra Universalis 53
2005
Later among the works it cites.
I.Chakir, M.Pouzet, The length of chains in algebraic modular lattices, ORDER 24
2007
Later among the works it cites.
I.Chakir, Conditions de chaînes dans les treillis algébriques, Université de Settat, Octobre 2007, 106 pp. fichier .pdf
2007
Later among the works it cites.

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