I'm wondering about the question



> 
> "If we have a finitely presented \_\_, is it necessarily finitely presented with respect to any finite generating set for it?" 
> 
> 
> 


I know this is true for groups and for [$R$-modules](https://mathoverflow.net/questions/1788/does-finitely-presented-mean-always-finitely-presented-answered-yes). Does anyone know whether this is true for $A$-algebras? Commutative $A$-algebras? Other things people might happen to know about it for?



Does "finitely presented" mean "always finitely presented", considered in general
Yes in general.


See Adámek and Rosicky, Locally Presentable and Accessible Categories Cambridge University Press, Cambridge, (1994).


T. 3.12 p. 143.


Of course "in general" I mean: every "algebraic theory" (many sorted) Set models. 


For topological algebraic structures, (like profinite groups) this equivalence isn't true for the Yiftach Barnea example 


