  The success of SQL, NoSQL, and NewSQL databases is a reflection of their
ability to provide significant functionality and performance benefits for
specific domains, such as financial transactions, internet search, and data
analysis. The BigDAWG polystore seeks to provide a mechanism to allow
applications to transparently achieve the benefits of diverse databases while
insulating applications from the details of these databases. Associative arrays
provide a common approach to the mathematics found in different databases: sets
(SQL), graphs (NoSQL), and matrices (NewSQL). This work presents the SQL
relational model in terms of associative arrays and identifies the key
mathematical properties that are preserved within SQL. These properties include
associativity, commutativity, distributivity, identities, annihilators, and
inverses. Performance measurements on distributivity and associativity show the
impact these properties can have on associative array operations. These results
demonstrate that associative arrays could provide a mathematical model for
polystores to optimize the exchange of data and execution queries.
