In the general sense of an algebra (a set with some operations, as in Universal Algebra courses), is it always possible to construct any full algebra (up to isomorphism) just from its finitely generated subalgebras, by the taking of suitable direct products?
2026-03-30 16:59:31.1774889971
Construct an algebra from its finitely generated algebras
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It is not possible to do it with direct product. There are many counterexamples, but I will take one that is right now on the top of my head: any atomless Boolean algebra is not a direct product of finite (that is the same as finitely generated) Boolean algebras, because every direct product of finite Boolean algebras is an atomic Boolean algebra.
However, every algebra is a direct limit of its finitely generated subalgebras.