The product of divisibility groups (G1,O1)×(G2,O2) with G1 finitely generated

We show that: (1) If (G1,O1) is a finitely generated divisibility group that is not lattice ordered and if (G2,O2) is a divisibility group such that Card(G2)>Card(R), then the product (G1,O1)×(G2,O2) is not a divisibility group.(2) If (G1,O1) is a torsion free finitely generated divisibility group and if (G2,O2) is a lattice ordered group with only finitely many ultra filters, then the product (G1,O1)×(G2,O2) is a divisibility group if and only if Card(G2)⩽Card(R).(3) If (G1,O1) and (G2,O2) are both torsion free finitely generated divisibility groups, then the product (G1,O1)×(G2,O2) is a divisibility group.