On cardinalities of row spaces of Boolean matrices |
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Authors: | Janusz Konieczny |
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Institution: | (1) Department of Mathematics, Pennsylvania State University, 16802 University Park, PA |
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Abstract: | We prove that there are exactlyn numbers greater than 2
n−1 that can serve as the cardinalities of row spaces ofn×n Boolean matrices. The numbers are: 2
n−1+1,2
n−1+2,2
n−1+4, ..., 2
n−1+2
n−2, 2
n
. Two consequences follow. The first is that the height of the partial order ofD-classes in the semigroup ofn×n Boolean matrices is at most 2
n−1+n−1. The second is that the numbers listed above are precisely the numbers greater than 2
n−1 that can serve as the cardinalities of topologies on a finite setX withn elements. |
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Keywords: | |
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