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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 $\vcone(P)$. |
672 (See Figure \nn{need figure}.) |
672 (See Figure \ref{vcone-fig}.) |
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673 \begin{figure}[t] |
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674 $$\mathfig{.65}{tempkw/vcone}$$ |
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675 \caption{(a) $P$, (b) $P\times I$, (c) $\Cone(P)$, (d) $\vcone(P)$}\label{vcone-fig} |
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676 \end{figure} |
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} |