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Is there a name for functions with the following property (a la transitive relations)?

If $F(X \cup \{a\}) = y$ and $F(X \cup \{b\}) = y$ then $F(X \cup \{a,b\}) = y$

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@David: I doubt it, but the well known union-closed set conjecture of Frankl deals with families of sets that are closed under unions. A function constant on such a family is similar to your function, so you might delve into that literature to see if anyone has coined a term. How did it come up? –  Eric Tressler May 8 '13 at 2:29
Maybe if you told us where this function comes from (at least what discipline) and what other properties it has then we might be able to help. Bear in mind that MathOverflow is an interdiciplinary site, so the more context and explanation you give, the more enlightening response you might get from someone with a different point of view. –  Paul Taylor May 8 '13 at 7:52
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