Let $A$ be the set of $2\times 2$ Boolean matrices with $MRN$ if and only if $m_{ij}\le n_{ij}$ for $i,j=1,2$. Is $(A,R)$ a lattice?

Question

A Diagraph is said to be a lattice if every pair in lattice has least upper bound (LUB) and Greatest Lower Bound (GLB).

Answer 1

In fact the way the different matrices should be arranged is as a 4D hypercube.

See the graphics below where:

$$(a,b,c,d) \ \ \ \ \leftrightarrow \ \ \ \ \begin{pmatrix}a&b\\c&d\end{pmatrix}$$

In fact, you don't need to represent this graphics. You need only say that:

$$\begin{cases}GLB((a,b,c,d),(a',b',c',d'))&=&(\min(a,a'),\min(b,b'),\min(c,c'),\min(d,d'))\\ LUB((a,b,c,d),(a',b',c',d'))&=&(\max(a,a'),\max(b,b'),\max(c,c'),\max(d,d'))\end{cases}$$

for example:

$$LUB((0101),(1000))=(1101)$$