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?
2026-04-07 19:02:09.1775588529
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...