Abstract
The Dilworth numbers of a finite partially ordered set $P,\{ d_k ( P )|k\geqq 0 \}$, are the maximum sizes of the union of k antichains. We give a characterization of the Dilworth numbers in terms of the dimensions of the kernels of certain linear maps on the complex vector space generated by the elements of P. This is applied to prove bounds on the Dilworth numbers of product partial orders in terms of those of its factors and to prove sufficient conditions for the product of two partial orders to have the Sperner property.
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