667 Before stating the axiom we need a few preliminary definitions.

   667 Before stating the axiom we need a few preliminary definitions.

   668 If $P$ is a poset let $P\times I$ denote the product poset, where $I = \{0, 1\}$ with ordering $0\le 1$.

   668 If $P$ is a poset let $P\times I$ denote the product poset, where $I = \{0, 1\}$ with ordering $0\le 1$.

   669 Let $\Cone(P)$ denote $P$ adjoined an additional object $v$ (the vertex of the cone) with $p\le v$ for all objects $p$ of $P$.

   669 Let $\Cone(P)$ denote $P$ adjoined an additional object $v$ (the vertex of the cone) with $p\le v$ for all objects $p$ of $P$.

   670 Finally, let $\vcone(P)$ denote $P\times I \cup \Cone(P)$, where we identify $P\times \{0\}$ with the base of the cone.

   670 Finally, let $\vcone(P)$ denote $P\times I \cup \Cone(P)$, where we identify $P\times \{0\}$ with the base of the cone.

   671 We call $P\times \{1\}$ the base of $\vcone(P)$.

   671 We call $P\times \{1\}$ the base of $\vcone(P)$.

   673 

   677 

   674 \nn{maybe call this ``splittings" instead of ``V-cones"?}

   678 \nn{maybe call this ``splittings" instead of ``V-cones"?}

   675 

   679 

   676 \begin{axiom}[V-cones]

   680 \begin{axiom}[V-cones]

   677 \label{axiom:vcones}

   681 \label{axiom:vcones}