Аннотация
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.