Canonical extensions and profinite limits in finitely generated varieties of lattice based algebras

The lattice reduct of any lattice based algebra A is embeddable into a compact, dense complete lattice, known as its canonical extension. Two ways of extending the additional operations of A have been considered, giving rise to algebras of the same type of A, the canonical extension of A and the dual canonical extension of A. In a finitely generated variety every algebra is embeddable into its profinite limit, which lattice reduct turns out to be a compact, dense complete lattice. This allows to show that the canonical extension and the dual canonical extension of an algebra A coincide and they are, up to isomorphism, the profinite limit of A.