Bounds on the k-dimension of Products of Special Posets
Trotter conjectured that dimP×Q≥dimP+dimQ−2 for all posets P and Q. To shed light on this, we study the k-dimension of products of finite orders. For k ∈ o(ln n), the value 2dimk(P)−dimk(P×P) is unbounded when P is an n-element antichain, and 2dim2(mP)−dim2(mP×mP) is unbounded when P is a fixed pose...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Springer Science+Business Media
2014
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| Online Access: | http://hdl.handle.net/1721.1/85844 https://orcid.org/0000-0003-1303-5598 |
| Summary: | Trotter conjectured that dimP×Q≥dimP+dimQ−2 for all posets P and Q. To shed light on this, we study the k-dimension of products of finite orders. For k ∈ o(ln n), the value 2dimk(P)−dimk(P×P) is unbounded when P is an n-element antichain, and 2dim2(mP)−dim2(mP×mP) is unbounded when P is a fixed poset with unique maximum and minimum. For products of the “standard” orders S m and S n of dimensions m and n, dimk(Sm×Sn)=m+n−min{2,k−2} . For higher-order products of “standard” orders, dim2(∏ti=1Sni)=∑ni if each n i ≥ t. |
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