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