Generalized relations
Date: Mon, 21 Dec 2015 11:26:29 -0800 (PST)
Message-ID: <07d179a7-c87a-4fc2-8664-a7ffe3c03f98_at_googlegroups.com>
In parallel (Norbert's topological databases) thread there have been touched a topic of constraint databases. Conventional wisdom is representing relations as semi-algebraic sets. The motivation for that is quite obvious: Tarski-Seidenberg theorem. Tarski-Seidenberg asserts that projection of semialgebraic set is semialgebraic. Together with "folklore knowledge" that the class of all semialgebraic subsets is closed under finite unions and intersections, taking complement, inverse image by a polynomial mapping, and cartesian product it becomes obvious that semialgebraic sets fit nicely into Codd's relational algebra.
Unlike semi-algebraic sets, algebraic sets (polynomial varieties) are not closed under projection. So why insist on their significance?
First, something has to be done about projection. One needs to treat projection of algebraic set as geometric projection, followed by elimination of variables. Such combined operation transforms agebraic set into algebraic set. After that one can see that algebra-geometry dictionary on page 214 of classic textbook http://www.math.ku.dk/~holm/download/ideals-varieties-and-algorithms.pdf hints what Codd's relational algebra of varieties (or ideals) is.
Little more coherent exposition of this idea: https://vadimtropashko.wordpress.com/2015/06/27/lattice-of-ideals/ However, please keep in mind that the assertion that radical ideal is an abstraction capturing database relation is wrong. We need to equip an ideal together with space it lives in, which makes the whole construction a little bit more clumsy.
Again, even with those difficulties I still stand by belief that algebraic theory is superior to semi-algebraic. For once, is there such ubiqutous algebra-geometry correspondence as in algebraic case? Received on Mon Dec 21 2015 - 20:26:29 CET