A (Successful?) commercial implementation of ExtendedSetTheory: "the A2DB advanced analytic database. A2DB employs patented Algebraix database technology based on advanced set-processing mathematics. By enabling real-time data access, eliminating manual performance tuning and running on affordable commodity hardware, Algebraix database technology promises to be disruptive to the entire business-intelligence (BI) complex."

The patent they talk about is: "U.S. Patent No. 7,613,734, for its systems and methods for providing data sets using a store of algebraic relations."

Their website: http://www.algebraixdata.com

The patent: http://www.patentstorm.us/patents/7613734/fulltext.html
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Can they really patent ExtendedSetTheory? Is it too much "math based" to be patentable?

''It appears they haven't patented ExtendedSetTheory itself, but have patented a particular application of it.''

So... now they are the only ones that can implement a database with it?

''No.  The patent does not claim to have invented ExtendedSetTheory even if it could patent a mathematical technique (which is attributed to D L Childs anyway), nor does it appear (having only glanced over it, not read it in detail) to patent applying ExtendedSetTheory to databases in general.  It appears to patent a specific strategy for implementing of ExtendedSetTheory for analytical databases.  You would only potentially run afoul of the patent if you implemented your ExtendedSetTheory database system the same way as described in the patent.  This is quite distinct from, say, the TransRelationalModel, in which the fundamental technique is the substance of the related patents.  I'm not aware of any way to implement the TransRelationalModel without running afoul of the patent, even though the TransRelationalModel might be an unintentional (or even intentional, but not stated as such) application of a particular sequence of ExtendedSetTheory operators.''

''It's notable that most of the techniques described in the patent obviously have prior art, so even if you implemented an ExtendedSetTheory database system the same way described in the patent, it's questionable as to whether it would be upheld in court.''