Identities and Quasi-Identities of Pointed Algebras

Each pointed enrichment of an algebra can be regarded asthe 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 algebrain a finitely axiomatizable residually very finite variety admits a finite basis of identities.We also prove thatevery pointed enrichment of a finite algebrain a directly representable quasivarietyadmits a finite basis of quasi-identities.We give some applications of these results and examples.