On generalizing free algebras for a functor

In this article we introduce a new setting, based on partial algebras, for studying constructions of finitely generated free algebras. We give sufficient conditions under which the finitely generated free algebras for a variety V may be described as the colimit of a chain of finite partial algebras obtained by repeated application of a functor. In particular, our method encompasses the construction of finitely generated free algebras for varieties of algebras for a functor as in Bezhanishvili and Kurz (2007, LNCS, 143–157), Heyting algebras as in Bezhanishvili and Gehrke (2011, LMCS, 7, 1–24) and S4 algebras as in Ghilardi (2010, J. Appl. Non-classical, Logics, 20, 193–217).