In the slides Provenance for Database Transformations, page 24, they provide a semiring for lineage, which include a $\cup^*$ symbol. However, I can not find any related materials about the meaning of this symbol. To be more specific, the elements in this semiring should be set of different symbols, e.g, $\{a,b,c\}$. Any help would be appreciated.
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2$\begingroup$ If you can find the paper cited two lines earlier on page 24, that should define $\cup^*$, because it's not a standard set-theoretic notation. Otherwise, you might ask Tannen by email. $\endgroup$– Andreas BlassCommented Dec 27, 2020 at 4:35
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3$\begingroup$ A wild guess: the bottom of the slide suggests that $\cup$ and $\cup^*$ are operations defining a semiring structure, which in particular means that one distributes over the other. This suggets that $\cup^*$ stands for intersection. The slide also seems to be saying that the neutral elements are $\emptyset$ and $\emptyset^*$. So perhaps the author writes $*$ for set-theoretic complement (relative to a ground set), so that $\cup^*$ denotes the de Morgan dual of $\cup$. $\endgroup$– Tobias FritzCommented Dec 27, 2020 at 7:40
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$\begingroup$ @AndreasBlass Yeah I checked that paper but didn't find the definition of this symbol. I guess in that paper they define the lineage and Tannen worked on this symbol for the semiring. $\endgroup$– BrandNewStoryCommented Dec 28, 2020 at 1:52
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$\begingroup$ @TobiasFritz I'm not sure if I understand you correctly. In that case, if given a•b, what should it be under ∪∗ operation? $\endgroup$– BrandNewStoryCommented Dec 28, 2020 at 1:56
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$\begingroup$ @BrandNewStory: now I don't know what you mean by $a\bullet b$. All I meant is that $\cup^*$ may stand for the intersection of sets, just as $\cup$ stands for the union of sets. That's my guess. $\endgroup$– Tobias FritzCommented Dec 28, 2020 at 7:00
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