Abstract
Given a maximal subchainC of a semilatticeS, there are some natural ‘leaves’ ofS attached to it. These are subsemilattices ofS which may have a simpler structure thanS itself. We look atS as build up fromC together with its leaves. Starting with one-point subsemilattices, the ‘(branching) rank’ ofS is defined to be the least number of steps needed to recoverS. For technical reasons, only semilattices with no infinite descending chains are considered. The main result states that ifR is a subsemilattice ofS and rankS is defined, then rankR≤rankS. On the other hand, rank does not behave well with respect to epimorphisms. Several examples are presented as well as various results concerning finite semilattices and trees.
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Communicated by M. Pouzet
This work was supported, in part, by NSERC Grant A4044.
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Almeida, J. A notion of branching rank for semilattices with descending chain condition. Order 4, 397–409 (1988). https://doi.org/10.1007/BF00714480
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DOI: https://doi.org/10.1007/BF00714480