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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Weighted join semilattices and transversal matroids


Author: Richard A. Brualdi
Journal: Trans. Amer. Math. Soc. 191 (1974), 317-328
MSC: Primary 05B35
DOI: https://doi.org/10.1090/S0002-9947-1974-0382039-7
MathSciNet review: 0382039
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Abstract: We investigate join-semilattices in which each element is assigned a nonnegative weight in a strictly increasing way. A join-subsemilattice of a Boolean lattice is weighted by cardinality, and we give a characterization of these in terms of the notion of a spread. The collection of flats with no coloops (isthmuses) of a matroid or pregeometry, partially ordered by set-theoretic inclusion, forms a join-semilattice which is weighted by rank. For transversal matroids these join-semilattices are isomorphic to join-subsemilattices of Boolean lattices. Using a previously obtained characterization of transversal matroids and results on weighted join-semilattices, we obtain another characterization of transversal matroids. The problem of constructing a transversal matroid whose join-semilattice of flats is isomorphic to a given join-subsemilattice of a Boolean lattice is then investigated.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0382039-7
Keywords: Weighted semilattice, spread, Boolean lattice, transversal matroid
Article copyright: © Copyright 1974 American Mathematical Society