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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)?

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What kind of object do you want the result to be and what kind of relation do you want it to bear to the original objects? – Qiaochu Yuan Mar 15 '12 at 18:21
I would like see if [idempotent] semiring, or relation algebra can be constructed this way from lattice and monoid. – Tegiri Nenashi Mar 15 '12 at 18:30
You can try to define them in terms of functors from a product category, but there is nothing within universal algebra that I know of that would let you do that. – Arturo Magidin Mar 15 '12 at 18:55
@Tegiri: You can view the construction as a functor from a subcategory of $\mathbf{Ring}\times\mathbf{Monoid}$ to $\mathbf{Ring}$ (the localization functor), but this is not a universal-algebra construction in the same sense as products, quotients, etc. (And that should be $p\in\mathbb{Z}$, $q\in\mathbb{Z}-\{0\}$). Precisely what I said: for specific instances, you can define such a construction in terms of functors from an appropriate product category, but I don't see how you can view this as a "general algebra" construction of algebras of different signatures. – Arturo Magidin Mar 15 '12 at 19:04
There's a generic definition of a "tensor product" 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. – Zhen Lin Mar 15 '12 at 19:57

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