Ultracritical and hypercritical binary structures

A binary structure is an arc-coloured complete digraph, without loops, and with exactly two coloured arcs (u,v)(u,v) and (v,u)(v,u) between distinct vertices uu and vv. Graphs, digraphs and partial orders are all examples of binary structures. Let BB be a binary structure. With each subset WW of the vertex set V(B)V(B) of BB we associate the binary substructure B[W]B[W] of BB induced by WW. A subset CC of V(B)V(B) is a clan of BB if for any c,d∈Cc,d∈C and v∈V(B)∖Cv∈V(B)∖C, the arcs (c,v)(c,v) and (d,v)(d,v) share the same colour and similarly for (v,c)(v,c) and (v,d)(v,d). For instance, the vertex set V(B)V(B), the empty set and any singleton subset of V(B)V(B) are clans of BB. They are called the trivial clans of BB. A binary structure is primitive if all its clans are trivial.With a primitive and infinite binary structure BB we associate a criticality digraph (in the sense of [11]) defined on V(B)V(B) as follows. Given v≠w∈V(B)v≠w∈V(B), (v,w)(v,w) is an arc of the criticality digraph of BB if vv belongs to a non-trivial clan of B[V(B)∖{w}]B[V(B)∖{w}]. A primitive and infinite binary structure BB is finitely critical if B[V(B)∖F]B[V(B)∖F] is not primitive for each finite and non-empty subset FF of V(B)V(B). A finitely critical binary structure BB is hypercritical if for every v∈V(B)v∈V(B), B[V(B)∖{v}]B[V(B)∖{v}] admits a non-trivial clan CC such that |V(B)∖C|≥3|V(B)∖C|≥3 which contains every non-trivial clan of B[V(B)∖{v}]B[V(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].