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

  • $\begingroup$ Welcome to MSE. Please include your question in the body of the question, instead of putting it only in the title. $\endgroup$ – José Carlos Santos Oct 17 at 9:54
  • $\begingroup$ What is your personal work ? There are 16 such matrices which is a small number (In fact, it could be even boolean vectors of size 4). Have you attempted to make a sketch of this (assumed) lattice ? $\endgroup$ – Jean Marie Oct 17 at 9:55
  • $\begingroup$ Thank You so much @JoséCarlosSantos .I will take care next time. $\endgroup$ – Diana Oct 17 at 10:45
  • $\begingroup$ @JeanMarie - I got the 16 such matrices . I don't have any idea about how to proceed ahead. $\endgroup$ – Diana Oct 17 at 10:51
  • $\begingroup$ Draw an arrow each time you have a relationship between them and see if every pair has a LUB and GLB. $\endgroup$ – Jean Marie Oct 17 at 10:57

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:


enter image description here

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    $\begingroup$ Thank you for the answer ,It explains all my doubts .Sorry for late reply $\endgroup$ – Diana Oct 19 at 19:36

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