Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 1999; 36(3): 633-648

Printed May 1, 1999

Copyright © The Korean Mathematical Society.

The dimension of the convolution of bipartite ordered Sets

Deok Rak Bae

Abstract

In this paper, for any two bipartite ordered sets $P$ and $Q,$ we define the
convolution $P*Q$ of $P$ and $Q.$ For $\dim(P)=s$ and $\dim(Q)=t,$ we prove
that
$s+t-(U+V)-2\le \dim(P*Q)\le s+t-(U+V)+2, $ where $U+V$ is the max-min
integer of the certain realizers. In particular, we also prove that
$\dim(P_{n,k})= n+k-\lfloor \frac{n+k}{3}\rfloor$
for $2\le k\le n<2k$
and $\dim(P_{n,k})= n$ for $n\ge 2k,$
where $P_{n,k}=S_n*S_k$ is
the convolution of two standard ordered sets $S_n$ and $S_k.$

Keywords: bipartite ordered set, dimension, Ferrers relation, Ferrers dimension, realizer, $R$-irreducible, max-min integer

MSC numbers: 06A07

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