Answer to some open questions on covering dimension for finite lattices

作者：Zhang, Hai-feng^*; Zhou, Meng; Zhang, Guang-jun

来源：Discrete Mathematics, 2017, 340(5): 1086-1091.

DOI：10.1016/j.disc.2016.10.019

• Lattice theory
• Covering dimension

Dube, Georgiou, Megaritis and Moshokoa (2015) characterized the covering dimension of a finite lattice. The covering dimension is completely different from the order dimension. The covering dimension of a finite lattice L will be denoted by dim(L) in this paper. Dube, Georgiou, Megaritis and Moshokoa (2015) posed three open questions. In this paper we answer two of these three questions. Question 1 is that whether the relation dim(L-1 x L-2) <= dim(L-1) dim(L-2) + 1 holds for all finite lattices L-1 and L-2. We give a positive answer for the above Question 1. Question 2 is that whether the relation dim(L(1)x L-2) <= dim(L-1) + dim(L-2) + 1 holds for all finite lattices L-1 and L-2. We give a negative answer to the above Question 2 by an example.

• 出版日期2017-5
• 单位北京航空航天大学