Not every splitting Heyting or interior algebra is finitely presentable (Q454375)

A pair \(V_1, V_2\) of subvarieties of a variety \(V\) is called a splitting pair if one of them is not a subvariety of the other and for every subvariety \(W\) of \(V\) either \(V_1\) or \(V_2\) is a subvariety of \(W\). It is known that, if \(V_1, V_2\) is a splitting pair, then the variety \(V_1\) is generated by a finitely generated subdirectly irreducible algebra; such an algebra is called a splitting algebra. On the other hand, the variety \(V_2\) can be defined by a single identity. The paper contains an example of a variety of Heyting algebras and of a splitting algebra in this variety which is not finitely presentable. It is shown that the corresponding splitting pair cannot be defined by any finitely presentable algebra. Using the Gödel-McKinsey-Tarski translation and the Blok-Esakia theorem, the author constructs a variety of Gregorczyk algebras with similar properties.