Canonical, natural and profinite extensions of semilattices

The work we present here was motivated by recent results obtained for lattice-based algebras in finitely generated varieties which reconcile canonical, natural and profinite extensions of those algebras. The variety of join-semilattices with 0 is generated, as a quasivariety, by the two-element semilattice. This guarantees the existence and coincidence of both natural and profinite extensions of any semilattice in the variety. Those extensions are complete lattices with some good properties regarding canonicity, considering now the extended notion of canonical extension of bounded lattices to posets. Canonical extensions of semilattices are obtained. In the particular case of lattices, their canonical extensions coincide with the canonical extensions of their semilattice reducts, providing us a new approach and a better understanding of canonicity.