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Are there any particular term for a pair $(U;A)$ where $U$ is a set and $A\in\mathscr{P}U$? That is, saying informally, $(U;A)$ is a set $A$ together with a set $U$ on which $A$ is defined ($A$ is defined as a subset of $U$).

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One could consider this pair as the inclusion $A\subset U$... – Simon Markett May 20 '12 at 20:05
How about extension? But I don't think there is a standard term for that. The closer I know of is pointed set. – lhf May 20 '12 at 20:45
You could call it a structure with one unary relation, or simply a unary relation $A$ on $U$. – Trevor Wilson Oct 12 '12 at 20:09

For more general situations of this kind, there are indeed names.

For example, one can talk about filtration, if you have a set of sets with certain inclusion properties.

If the sets are vector spaces, this is a (very trivial) flag.

But normally, one does not use this terminology for the simple situation of "a set and a subset". You can view it as an element of the inclusion relation, but there are few contexts where this would be helpful.

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