Axiomatization of CG ∘ S by PC plus qcg

Determine whether the quasivariety CG ∘ S, where CG is the variety of commutative groupoids and S is the variety equivalent to semilattices, is axiomatized by the union of (i) the family of identities PC consisting of all identities of the form (x1…xm r1)·(x1…xm r2) = (x1…xm r2)·(x1…xm r1) for every m > 0 and r1, r2 ∈ Tm, and (ii) the quasi-identity qcg: (zx = x ∧ zy = y ∧ xz = yz) → (xy = yx).

Background

The paper studies Mal'tsev products V ∘ S, with S the variety equivalent to semilattices, and conditions under which V ∘ S is a variety. For the regular variety CG of commutative groupoids, it is shown that CG ∘ S is not a variety, but certain algebraic consequences hold.

They define the prolongation #1{CG} via the identity family PC: all identities (x1…xm r1)·(x1…xm r2) = (x1…xm r2)·(x1…xm r1) for m > 0 and r1, r2 ∈ Tm. Proposition 4.4 proves that CG ∘ S satisfies the quasi-identity qcg: (zx = x ∧ zy = y ∧ xz = yz) → (xy = yx). The open question asks whether these two pieces—PC and qcg—suffice to axiomatize CG ∘ S.