Identities and quasi-identities of pointed algebras

Each pointed enrichment of an algebra can be regarded as the same algebra with an additional finite set of constant operations. An algebra is pointed whenever it is a pointed enrichment of some algebra. We show that each pointed enrichment of a finite algebra in a finitely axiomatizable residually very finite variety admits a finite basis of identities. We also prove that every pointed enrichment of a finite algebra in a directly representable quasivariety admits a finite basis of quasi-identities. We give some applications of these results and examples.