Stephen B. Maurer

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A combinatorial pregeometry, or matroid, may be defined as a finite set of elements E and a collection of bases M, all subsets of E, such that for all B,B' G âiï and any é eB' — £, there exists e e B — B' for which B — e + e' e&. This exchange axiom suggests it is fruitful to represent a pregeometry Jt by a graph : Let there be a vertex for each basis and(More)
In this paper, we extend the notion of a circulant to a broader class of vertex.transitive graphs, which we call multidimensional circulants. This new class of graphs is shown to consist precisely of those vertex-transitive graphs with an automorphism group containing a regular abelian subgroup. The result is proved using a theorem of Sabidussi which shows(More)
The rank (resp. dimension) of a poset P is the cardinality of a largest (resp. smallest) set of linear orders such that its intersection is P and no proper subset has intersectior: P. Dimension has been studied extensively. Rank was introduced recently by Maurer and Rabinovitch in [4], where the rank of antichains was determined. In this paper we develop a(More)
The size of a minimum realizer of a partial order is called the dimension of that partial order. Here we initiate the study of minimal realizers which are not minimum. As an aid to the study of such realizers, we associate to each minimal realizer certain critical digraphs. We characterize all such critical digraphs for the antichain on n elements, and(More)