Union of two partial orderings
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As exitingcorpse remarks, antisymmetry may fail.
Transitivity can be a problem too, for example on domain $\{a,b,c\}$, with $S=\{(a,a), (b,b), (a,b),(c,c)\}, R= \{(a,a),(b,c),(c,c),(b,b)\}$. Then the union $S \cup R$ is not transitive: $(a,c)$ should be in it as $(a,b)$ and $(b,c)$ are...
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Heisenberg
Updated on June 11, 2020Comments
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Heisenberg over 3 years
Suppose S and R are partial orderings. Does is necessarily mean that $R \cup S$ (union) is a partial ordering? If not what conditions would have to be met for it to be a partial ordering?
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Cameron Buie over 10 yearsA sufficient (but perhaps not necessary) condition is that the domains of $S$ and $R$ be disjoint.
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citedcorpse over 10 yearstake the set to be $\{a,b\}$ with $S = \{(a,a), (a,b), (b,b)\}$ and $R = \{(a,a),(b,a),(b,b)\}$. Then the union fails to be antisymmetric.
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Git Gud over 10 years
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Lord_Farin over 10 years