For which varieties $\mathbf{V}$ (in the sense of universal algebra) do the finite algebras in $\mathbf{V}$ form a finitely cocomplete category, or more strongly, are closed under finite colimits in $\mathbf{V}$?
Some examples:
- Sets
- (Left or right) $M$-sets where $M$ is any monoid
- Commutative (or abelian) semigroups
- Commutative (or abelian) monoids
- Abelian groups
- (Left or right) $R$-(semi)modules where $R$ is any ring (or rig)
- Boolean algebras
Some non-examples:
- Semigroups, monoids, groups, or rings (not necessarily commutative)
- Commutative rings (although the tensor product (coproduct) of two finite commutative rings is finite, the initial ring $\mathbf{Z}$ is infinite)