TITLE: Equational theories with the same variables on both sides
QUESTION [5 upvotes]: I'm interested in classes of algebraic structures defined by sets of equations in which the same variables occur on both sides of every equation; e.g., the class of semilattices, defined by the associativity, commutativity, and idempotence equations. (Is there a name for such varieties?) In particular, is the class of Boolean algebras definable by a set of such equations, and, if so, is the class of lattices? (I know BAs are if lattices are.)

REPLY [5 votes]: Is there a name for such varieties?
An identity which has the same variables on both sides is called regular. Varieties defined by regular identities are sometime also called regular, although "regular variety" sometimes means a different thing. I think that it was Plonka who initiated the study of regular identities:
Plonka, J.
On equational classes of abstract algebras defined by regular equations.
Fund. Math. 64 1969 241–247. 
It is known that a variety in an algebraic language $L$ is axiomatizable by regular identities iff the variety contains a $2$-element "$L$-semilattice" (which is the type of structure bof used in his proposition).