Equational theories with the same variables on both sides

Question

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.)

Answer 1

For convenience, let me call an equation "good" if the same variables occur on both sides, "bad" otherwise.

The answer to your question is negative for lattices and for Boolean algebras. More generally:

Proposition. Suppose $K$ is an equational class defined by a set of good equations. Then every equation which holds identically in all members of $K$ is good.

Proof. We consider algebraic structures of a fixed signature. It will suffice to exhibit a structure $\mathfrak A=\langle A,F_1,F_2,\dots\rangle$ such that every good equation, but no bad equation, holds identically in $\mathfrak A.$ To this end, let $A=\{0,1\},$ and for each $i$ define $$F_i(x_1,x_2,\dots,x_{n_i})=\prod_{k=1}^{n_i}x_k$$ It follows that any term $t$ evaluates to $1$ if all variables occurring in $t$ are assigned the value $1,$ and evaluates to $0$ otherwise. Hence, if the same variables occur on both sides of an equation $t_1=t_2,$ the equation will hold for any assignment of values to the variables. On the other hand, if some variable $x$ occurs on only one side of the equation, we can make the equation fail by assigning the value $0$ to $x$ and the value $1$ to all other variables.

Answer 2

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).