Ultracritical and hypercritical binary structures

Abstract

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 $V\left(B\right)$ of $B$ we associate the binary substructure $B\left[W\right]$ of $B$ induced by $W$. A subset $C$ of $V\left(B\right)$ is a clan of $B$ if for any $c,d\in C$ and $v\in V\left(B\right)\setminus C$, the arcs $(c,v)$ and $(d,v)$ share the same colour and similarly for $(v,c)$ and $(v,d)$. For instance, the vertex set $V\left(B\right)$, the empty set and any singleton subset of $V\left(B\right)$ 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 $V\left(B\right)$ as follows. Given $v\ne w\in V\left(B\right)$, $(v,w)$ is an arc of the criticality digraph of $B$ if $v$ belongs to a non-trivial clan of $B[V\left(B\right)\setminus \left\{w\right\}]$. A primitive and infinite binary structure $B$ is finitely critical if $B[V\left(B\right)\setminus F]$ is not primitive for each finite and non-empty subset $F$ of $V\left(B\right)$. A finitely critical binary structure $B$ is hypercritical if for every $v\in V\left(B\right)$, $B[V\left(B\right)\setminus \left\{v\right\}]$ admits a non-trivial clan $C$ such that $|V\left(B\right)\setminus C|\ge 3$ which contains every non-trivial clan of $B[V\left(B\right)\setminus \left\{v\right\}]$. 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].