Prelinear Algebras in Relatively Regular Quasivarieties

Given a quasivariety of lattice-ordered algebras, the linearly ordered algebras therein generate the subquasivariety of prelinear algebras. In the case that there exist a constant 1 and binary term i such that the quasivariety satisfies: $1 leq i(x,y) Leftrightarrow x leq y$ , we give an explicit axiomatization of the prelinear subquasivariety, relative to the original quasivariety. The existence of 1 and i with the above property is equivalent to the quasivariety being ‘relatively 1^?-regular’, by which we mean that each relative congruence is characterized by the negative cone of its 1-class. Dual results hold in the positive cone case.