Idempotent Algebras and Symmetry

The quasivariety consisting of all the algebras of a given type that can be embedded as a subalgebra into some algebra which has a transitive automorphism group contains the variety of all idempotent algebras of the given type: every idempotent algebra B can be embedded as a retract into an algebra which has a transitive automorphism group and which is simultaneously a subdirect power of B and a direct limit of powers of B. This result applies in particular to lattices, bands, Steiner quasigroups and so on