Critically Twin Primitive 2-Structures

Abstract

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 $$\sigma $$? be a 2-structure. A subset $$C$$C of $$V(\sigma )$$V(?) is a clan of $$\sigma $$? if for each $$v\in V(\sigma ){\setminus }C$$v?V(?)\C, $$v$$v is linked in the same way to all the elements of $$C$$C. A 2-structure $$\sigma $$? is clan primitive if $$|V(\sigma )|\ge 3$$|V(?)|?3 and all its clans are trivial. A clan primitive 2-structure $$\sigma $$? is critically primitive if $$\sigma [V(\sigma ){\setminus }\{v\}]$$?[V(?)\{v}] is not primitive for every $$v\in V(\sigma )$$v?V(?). A clan of cardinality 2 is a twin pair. A 2-structure $$\sigma $$? is twin primitive if $$|V(\sigma )|\ge 3$$|V(?)|?3 and $$\sigma $$? has no twin pairs. A twin primitive 2-structure $$\sigma $$? is critically twin primitive if $$\sigma [V(\sigma ){\setminus }\{v\}]$$?[V(?)\{v}] is not twin primitive for every $$v\in V(\sigma )$$v?V(?). 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.