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On an axiom concerning the theory of “interval orders”. (Italian. English summary) Zbl 0717.06001
Summary: Given a set X with a strong order, satisfying the usual axioms \(a\nless a \forall a\) and transitivity, if we add the axiom: \((a_ 1\prec a_ 2)\wedge (b_ 1\prec b_ 2)\Rightarrow (a_ 1\prec b_ 2)\vee (b_ 1\prec a_ 2)\) we obtain the so called “interval order” which was deeply studied first by P. C. Fishburn. If we substitute this axiom with the following: \((a_ 1\prec a_ 2)\wedge (b_ 1\prec b_ 2)\Rightarrow (a_ 1\prec b_ 1)\vee (b_ 1\prec a_ 1)\vee (a_ 2\prec b_ 2)\vee (b_ 2\prec a_ 2),\) we obtain a different order structure which implies the “interval order”. Such order structure is analyzed and many results are obtained including a very simple proof of a theorem of representability analogous to a fundamental theorem on interval orders due to Fishburn.

MSC:
06A06 Partial orders, general
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