Is it possible to define a "direct product" of two algebras with different signatures? For example, boolean lattice $\otimes$ monoid? Perhaps we need to take some quotient to make sense of it (e.g Q is direct product of two groups with standard congruence)?
direct product of different algebras?
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abstract-algebra
universal-algebra
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1There's a generic definition of a "tensor prod$u$ct" of two monads, which algebraically amounts to the theory obtained by taking the disjoint union of the two signatures, the disjoint union of the axioms, and adding a suitable commutativity axiom for each pair of operations from the two different signatures. There are also weaker notions. – 2012-03-15