Critically Twin Primitive 2-Structures

Unlike graphs, digraphs or binary relational structures, the 2-structures do not define precise links between vertices. They only yield an equivalence between links. Also 2-structures provide a suitable generalization in the framework of clan decomposition. Let σ be a 2-structure. A subset C of V (σ) is a clan of σ if for each v ∈ V (σ) ∖ C, v is linked in the same way to all the elements of C. A 2-structure σ is clan primitive if ∣V (σ)∣ ≥ 3 and all its clans are trivial. A clan primitive 2-structure σ is critically primitive if σ[V (σ) ∖ {v}] is not primitive for every v ∈ V (σ). A clan of cardinality 2 is a twin pair. A 2-structure σ is twin primitive if ∣V (σ)∣ ≥ 3 and σ has no twin pairs. A twin primitive 2-structure σ is critically twin primitive if σ[V (σ) ∖ {v}] is not twin primitive for every v ∈ V (σ). First, a twin decomposition is provided for 2-structures. Second, twin primitivity is studied by proving analogues of results on clan primitivity. Third, the notion of critically closed subset is introduced to characterize critically twin primitive 2-structures. Mathematics Subject Classifications (2010): 05C75, 05C20