673 Let $\Cone(P)$ denote $P$ adjoined an additional object $v$ (the vertex of the cone) with $p\le v$ for all objects $p$ of $P$. |
673 Let $\Cone(P)$ denote $P$ adjoined an additional object $v$ (the vertex of the cone) with $p\le v$ for all objects $p$ of $P$. |
674 Finally, let $\vcone(P)$ denote $P\times I \cup \Cone(P)$, where we identify $P\times \{0\}$ with the base of the cone. |
674 Finally, let $\vcone(P)$ denote $P\times I \cup \Cone(P)$, where we identify $P\times \{0\}$ with the base of the cone. |
675 We call $P\times \{1\}$ the base of $\vcone(P)$. |
675 We call $P\times \{1\}$ the base of $\vcone(P)$. |
676 (See Figure \ref{vcone-fig}.) |
676 (See Figure \ref{vcone-fig}.) |
677 \begin{figure}[t] |
677 \begin{figure}[t] |
678 $$\mathfig{.65}{tempkw/vcone}$$ |
678 \centering |
679 \caption{(a) $P$, (b) $P\times I$, (c) $\Cone(P)$, (d) $\vcone(P)$}\label{vcone-fig} |
679 \begin{tikzpicture} |
|
680 [kw node/.style={circle,fill=orange!70}, |
|
681 kw arrow/.style={-latex, very thick, blue!70}, |
|
682 kw label/.style={cca}, |
|
683 ] |
|
684 |
|
685 \definecolor{cca}{rgb}{.1,.4,.3}; |
|
686 |
|
687 \node at (0,0) { |
|
688 \begin{tikzpicture} |
|
689 \draw |
|
690 (0,0) node[kw node](p1){} |
|
691 (1,.5) node[kw node](p2){} |
|
692 (2,0) node[kw node](p3){}; |
|
693 |
|
694 \draw[kw arrow] (p1) -- (p3); |
|
695 \draw[kw arrow] (p2) -- (p3); |
|
696 \draw[kw arrow] (p1) -- (p2); |
|
697 |
|
698 \draw[kw label] (1,-.6) node{(a)}; |
|
699 \end{tikzpicture} |
|
700 }; |
|
701 |
|
702 \node at (7,0) { |
|
703 \begin{tikzpicture} |
|
704 \draw |
|
705 (0,0) node[kw node](p1){} |
|
706 ++(0,2.5) node[kw node](q1){} |
|
707 (1,.5) node[kw node](p2){} |
|
708 ++(0,2.5) node[kw node](q2){} |
|
709 (2,0) node[kw node](p3){} |
|
710 ++(0,2.5) node[kw node](q3){} |
|
711 ; |
|
712 |
|
713 \draw[kw arrow] (p1) -- (p3); |
|
714 \draw[kw arrow] (p2) -- (p3); |
|
715 \draw[kw arrow] (p1) -- (p2); |
|
716 \draw[kw arrow] (q1) -- (q3); |
|
717 \draw[kw arrow] (q2) -- (q3); |
|
718 \draw[kw arrow] (q1) -- (q2); |
|
719 \draw[kw arrow] (p1) -- (q1); |
|
720 \draw[kw arrow] (p2) -- (q2); |
|
721 \draw[kw arrow] (p3) -- (q3); |
|
722 |
|
723 \draw[kw label] (1,-.6) node{(b)}; |
|
724 \end{tikzpicture} |
|
725 }; |
|
726 |
|
727 \node at (0,-5) { |
|
728 \begin{tikzpicture} |
|
729 \draw |
|
730 (0,0) node[kw node](p1){} |
|
731 (1,.5) node[kw node](p2){} |
|
732 ++(0,2.5) node[kw node](v){} |
|
733 (2,0) node[kw node](p3){} |
|
734 ; |
|
735 |
|
736 \draw[kw arrow] (p1) -- (p3); |
|
737 \draw[kw arrow] (p2) -- (p3); |
|
738 \draw[kw arrow] (p1) -- (p2); |
|
739 \draw[kw arrow] (p1) -- (v); |
|
740 \draw[kw arrow] (p2) -- (v); |
|
741 \draw[kw arrow] (p3) -- (v); |
|
742 |
|
743 \draw[kw label] (1,-.6) node{(c)}; |
|
744 \end{tikzpicture} |
|
745 }; |
|
746 |
|
747 \node at (7,-5) { |
|
748 \begin{tikzpicture} |
|
749 \draw |
|
750 (0,0) node[kw node](p1){} |
|
751 ++(-2,2.5) node[kw node](q1){} |
|
752 (1,.5) node[kw node](p2){} |
|
753 ++(-2,2.5) node[kw node](q2){} |
|
754 ++(4,0) node[kw node](v){} |
|
755 (2,0) node[kw node](p3){} |
|
756 ++(-2,2.5) node[kw node](q3){} |
|
757 ; |
|
758 |
|
759 \draw[kw arrow] (p1) -- (p3); |
|
760 \draw[kw arrow] (p2) -- (p3); |
|
761 \draw[kw arrow] (p1) -- (p2); |
|
762 \draw[kw arrow] (p1) -- (v); |
|
763 \draw[kw arrow] (p2) -- (v); |
|
764 \draw[kw arrow] (p3) -- (v); |
|
765 \draw[kw arrow] (q1) -- (q3); |
|
766 \draw[kw arrow] (q2) -- (q3); |
|
767 \draw[kw arrow] (q1) -- (q2); |
|
768 \draw[kw arrow] (p1) -- (q1); |
|
769 \draw[kw arrow] (p2) -- (q2); |
|
770 \draw[kw arrow] (p3) -- (q3); |
|
771 |
|
772 \draw[kw label] (1,-.6) node{(d)}; |
|
773 \end{tikzpicture} |
|
774 }; |
|
775 |
|
776 \end{tikzpicture} |
|
777 \caption{(a) $P$, (b) $P\times I$, (c) $\Cone(P)$, (d) $\vcone(P)$} |
|
778 \label{vcone-fig} |
680 \end{figure} |
779 \end{figure} |
681 |
780 |
682 \nn{maybe call this ``splittings" instead of ``V-cones"?} |
781 \nn{maybe call this ``splittings" instead of ``V-cones"?} |
683 |
782 |
684 \begin{axiom}[V-cones] |
783 \begin{axiom}[V-cones] |