一致数え上げ

$ \forall n\in\N と任意の半順序集合$ U に対して、全単射な単調写像$ \varphi:\underline U⤖[1..n] を一致数え上げと呼ぶ

$ [a..b]:=[a,b]\cap\N とした

$ [1..n] の順序は、自然数の順序とする

有限集合の定義より、一致数え上げがある半順序集合は必ず有限順序集合となる

名前の意味

$ \le_Uの並びに反しないように$ \underline Uの要素と自然数とを対応付けている

$ \le_\bullet：$ \bulletの順序

つまり、$ \le_Uの並びに一致させて数え上げるということ

例

このHasse図で定義される順序集合に対して、$ \begin{array}{c|cccccc}u&1&2&3&4&5&6\\\hline \varphi(u)&a&d&b&e&c&f\end{array}で定義される$ \varphiは一致数え上げになる

https://scrapbox.io/files/6602438552ce9d0024d0b862.svg

$ \varphiの順に赤くなるようにした

code:numbering.tikz(tex)

\usetikzlibrary{animations}

\begin{document}

\begin{tikzpicture}domain=0:4

\tikzset{

every node/.style={circle,draw,inner sep=2pt,fill=white,font=\Large},

every path/.style={very thick},

animate={

a:fill={0s="white",1s={jump,"transparent"},7s={jump,"red"},repeats},

a:draw={0s="white",1s={jump,"white"} ,7s={jump,"red"},repeats},

d:fill={0s="white",2s={jump,"transparent"},7s={jump,"red"},repeats},

d:draw={0s="white",2s={jump,"white"} ,7s={jump,"red"},repeats},

b:fill={0s="white",3s={jump,"transparent"},7s={jump,"red"},repeats},

b:draw={0s="white",3s={jump,"white"} ,7s={jump,"red"},repeats},

e:fill={0s="white",4s={jump,"transparent"},7s={jump,"red"},repeats},

e:draw={0s="white",4s={jump,"white"} ,7s={jump,"red"},repeats},

c:fill={0s="white",5s={jump,"transparent"},7s={jump,"red"},repeats},

c:draw={0s="white",5s={jump,"white"} ,7s={jump,"red"},repeats},

f:fill={0s="white",6s={jump,"transparent"},7s={jump,"red"},repeats},

f:draw={0s="white",6s={jump,"white"} ,7s={jump,"red"},repeats},

}

\def\s{2};

\coordinate (up) at (\s*2^-0.5,\s*2^-0.5);

\coordinate (down) at (\s*2^-0.5,\s*-2^-0.5);

\draw (0,0) nodelabel={right:$a$}(a){}

-- ++(up) nodelabel={right:$b$}(b){}

-- ++(up) nodelabel={right:$c$}(c){}

-- ++(down) nodelabel={right:$e$}(e){}

-- ++(up) nodelabel={right:$f$}(f){};

\draw (b)

-- ++(down) nodelabel={right:$d$}(d){}

-- (e);

\end{tikzpicture}

\end{document}