“…A code property, or independence, [15], is a set P of languages for which there is n ∈ N ∪ {ℵ 0 } such that L ∈ P, if and only if L ∈ P, for all L ⊆ L with 0 < |L | < n. If L is in P then we say that L satisfies P. Thus, L satisfies P exactly when all nonempty subsets of L with less than n elements satisfy P. A language L ∈ P is called P-maximal, or a maximal P code, if L ∪ {w} / ∈ P for any word w / ∈ L. We note that every L satisfying P is included in a maximal P code [15]. As far as we know, all code related properties in the literature [4,6,8,11,15,23,28] are code properties as defined here. The focus of this work is on 3-independences that can also be viewed as independences with respect to a binary relation in the sense of [29].…”