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 B [ W ] 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. 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 B [ V ( B ) ∖ { w } ] . A primitive and infinite binary structure B is finitely critical if B [ V ( 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 ) , B [ V ( B ) ∖ { v } ] admits a non-trivial clan C such that | V ( B ) ∖ C | ≥ 3 which contains every non-trivial clan of B [ V ( B ) ∖ { v } ] . A hypercritical binary structure is ultracritical whenever its criticality digraph is connected. tracritical 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].