Ultracritical and hypercritical binary structures

A binary structure is an arc-coloured complete digraph, without loops, and with exactly two coloured arcs (u,v) and (v,u) between distinct vertices u and v. Graphs, digraphs and partial orders are all examples of binary structures. Let B be a binary structure. With each subset W of the vertex set V(B) of B we associate the binary substructure BW] of B induced by W. A subset C of V(B) is a clan of B if for any c,d∈C and v∈V(B)?C, the arcs (c,v) and (d,v) share the same colour and similarly for (v,c) and (v,d). For instance, the vertex set V(B), the empty set and any singleton subset of V(B) are clans of B. They are called the trivial clans of B. A binary structure is primitive if all its clans are trivial.With a primitive and infinite binary structure B we associate a criticality digraph (in the sense of 11]) defined on V(B) as follows. Given v≠w∈V(B), (v,w) is an arc of the criticality digraph of B if v belongs to a non-trivial clan of BV(B)?{w}]. A primitive and infinite binary structure B is finitely critical if BV(B)?F] is not primitive for each finite and non-empty subset F of V(B). A finitely critical binary structure B is hypercritical if for every v∈V(B), BV(B)?{v}] admits a non-trivial clan C such that |V(B)?C|≥3 which contains every non-trivial clan of BV(B)?{v}]. A hypercritical binary structure is ultracritical whenever its criticality digraph is connected.The ultracritical binary structures are studied from their criticality digraphs. Then a characterization of the non-ultracritical but hypercritical binary structures is obtained, using the generalized quotient construction originally introduced in 1].