Order connectivity preserving families

The order connectivity preserving families are precisely those families whose comparable connected components and dependent connected components are equal. This implies that order connectivity preserving families generalize preorder containment families and irreducible containment families which are two of the major ways of converting partial orders into set systems. Here are some examples of such order connectivity preserving families:

#{#{0 1} #{2 3}}
#{#{} #{0} #{1} #{0 1}}
#{#{0} #{0 1} #{0 2} #{0 1 2 3}}

An example of a family of sets that is not order connectivity preserving is #{#{0 1} #{1 2}}. The two sets #{0 1} a #{1 2} are dependent despite being incomparable so this is not an order connectivity preserving family. Well it is possible to have a family of sets with such intersecting elements that is order connectivity preserving such as #{#{0 1} #{1 2} #{0 1 2}} considering that all uniquely order connected families preserve order connectivity it is only the laminar families which forbid dependent incomparable sets that are order connectivity preserving for all their subsets. All order connectivity preserving multichain families are laminar as is the case with multichain containment families.