Answer to some open questions on covering dimension for finite lattices

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(L1×L2)≤dim(L1)+dim(L2)+1 holds for all finite lattices L1 and L2. We give a positive answer for the above Question 1. Question 2 is that whether the relation dim(L1∘L2)≤dim(L1)+dim(L2)+1 holds for all finite lattices L1 and L2. We give a negative answer to the above Question 2 by an example.