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What is the standard name for an algebraic system that has the following operations and axioms: 1. Boolean algebra structure 2. a product that behaves like a cartesian product I am thinking here of a collection of sets that is a Boolean algebra and is closed under cartesian product. Thanks, jlouis P.S.: In general, I an looking for the standard names of algebraic systems that are of use when one wants to study a collection of sets that is a boolean algebra and that may be closed under cartesian product, or under a power set like operation, ....

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It is a kind of Boolean algebra with operators, with a specific binary operation, the cartesian product, which is indeed additive w.r.t. union ($(A\cup B)\times C=(A\times C) \cup( B\times C)$).

Perhaps you might also be interested in categories with cartesian products or cartesian closed categories.

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