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If we represent a relation using a logical matrix (so that if (a,b) is present in the relation then the element at row a and column b in the matrix is a 1), is there any way to determine that this matrix represents a partial order using matrix operations?

For example, if we have the relation {(1,1),(2,2),(3,3),(1,2)} represented by this matrix:

[tex]

\left( \begin{array}{ccc}

1 & 1 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1 \end{array} \right)

[/tex]

Is there some way, using matrix operations, to determine if this represents a partial order?

For example, if we have the relation {(1,1),(2,2),(3,3),(1,2)} represented by this matrix:

[tex]

\left( \begin{array}{ccc}

1 & 1 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1 \end{array} \right)

[/tex]

Is there some way, using matrix operations, to determine if this represents a partial order?

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