Pure Skew Lattices in Rings

Given a ring $R$, let $Ssubseteq R$ be a pure multiplicative band that isclosed under the cubic join operation $xnabla y = x+y+yx-xyx-yxy.$ We show that$left( S,cdot,nablaright) $ forms a pure skew lattice if and only if $S$satisfies the polynomial identity $left(xy-yxright)^{2}z = zleft(xy-yxright)^{2}$.We also examine properties of pure skew latticesin rings.