A proof on interval order [on hold]











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Let $X$ be a nonempty finite set and $≻$ a binary relation on X. We say that $≻$ is an interval order if $x≻y$ and $x'≻y'$ imply either $x≻y'$ or $x'≻y$, for every $x$,$y$,$x'$ and $y'$ in $X$.



Prove that $≻$ is an interval order iff there exist two real functions $f$ and $g$ on $X$ such that $x≻y$ iff $f(x)>g(y)$ for every $x$ and $y$ in $X$.






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put on hold as off-topic by Asaf Karagila♦ 14 hours ago


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    Let $X$ be a nonempty finite set and $≻$ a binary relation on X. We say that $≻$ is an interval order if $x≻y$ and $x'≻y'$ imply either $x≻y'$ or $x'≻y$, for every $x$,$y$,$x'$ and $y'$ in $X$.



    Prove that $≻$ is an interval order iff there exist two real functions $f$ and $g$ on $X$ such that $x≻y$ iff $f(x)>g(y)$ for every $x$ and $y$ in $X$.






    order-theory







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    qwert3 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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    put on hold as off-topic by Asaf Karagila♦ 14 hours ago


    This question appears to be off-topic. The users who voted to close gave this specific reason:


    • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Asaf Karagila

    If this question can be reworded to fit the rules in the help center, please edit the question.















      up vote
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      up vote
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      down vote

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      1





      Let $X$ be a nonempty finite set and $≻$ a binary relation on X. We say that $≻$ is an interval order if $x≻y$ and $x'≻y'$ imply either $x≻y'$ or $x'≻y$, for every $x$,$y$,$x'$ and $y'$ in $X$.



      Prove that $≻$ is an interval order iff there exist two real functions $f$ and $g$ on $X$ such that $x≻y$ iff $f(x)>g(y)$ for every $x$ and $y$ in $X$.






      order-theory







      share|cite|improve this question









      New contributor




      qwert3 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      Let $X$ be a nonempty finite set and $≻$ a binary relation on X. We say that $≻$ is an interval order if $x≻y$ and $x'≻y'$ imply either $x≻y'$ or $x'≻y$, for every $x$,$y$,$x'$ and $y'$ in $X$.



      Prove that $≻$ is an interval order iff there exist two real functions $f$ and $g$ on $X$ such that $x≻y$ iff $f(x)>g(y)$ for every $x$ and $y$ in $X$.






      order-theory




      order-theory






      share|cite|improve this question









      New contributor




      qwert3 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question









      New contributor




      qwert3 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question








      edited 13 hours ago









      Andrés E. Caicedo

      64.1k8157243




      64.1k8157243






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      asked 16 hours ago









      qwert3

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      New contributor





      qwert3 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      put on hold as off-topic by Asaf Karagila♦ 14 hours ago


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Asaf Karagila

      If this question can be reworded to fit the rules in the help center, please edit the question.




      put on hold as off-topic by Asaf Karagila♦ 14 hours ago


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Asaf Karagila

      If this question can be reworded to fit the rules in the help center, please edit the question.



























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