In their paper [43] Rota and Metropolis considered the partially ordered set F _n of all nonempty faces of the n-cube [0, 1]ⁿ for each n = 1, 2,…, equipped with the following operation: (⊔) taking the supremum A⊔ B of any two faces A and B of F _n , together with the following two partially defined operations: (⊓) taking the set-theoretic intersection A ⊓ B of any two intersecting faces A and B of F _n , and (Δ) when a face A is contained in another face B, taking the antipode Δ (B, A) of A in B.