Answer to some open questions on covering dimension for finite lattices

Abstract

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 $dim\left(L\right)$ 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 $dim(L_1\times L_2)\le dim\left(L_1\right)+dim\left(L_2\right)+1$ 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 $dim(L_1\circ L_2)\le dim\left(L_1\right)+dim\left(L_2\right)+1$ holds for all finite lattices $L_1$ and $L_2$. We give a negative answer to the above Question 2 by an example.