Can lattices which are also linear orders be characterized equationally?

Question

There is a definition of a lattice as an algebraic structure of type $\langle 2,2 \rangle$ which satisfies commutativity and associativity for both operations and the absorption laws connecting the two operations. We can define a partial order based on those operations, which gives the definition of a lattice as one kind of partial order. Can the class of lattices which are also linear orders be defined in terms of equations only? They can certainly be defined in terms of a first-order formula based on the two operations, but I am asking if they can be defined equationally.

Answer 1

No, since any equationally-defined class is closed under direct products. The product of two linearly-ordered lattices is not in general linearly ordered. (Note that this is exactly the same observation used to answer your other question.)