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Tolerances, interval orders, and semiorders. (English) Zbl 0809.06001
The concept of an interval order was introduced by P. C. Fishburn as an irreflexive relation $$P$$ on a set $$X$$ satisfying the so-called interval- order condition: if $$xPy$$ and $$zPw$$, then $$xPw$$ or $$zPy$$. A semiorder is an interval order $$P$$ satisfying: if $$xPy$$ and $$yPz$$, then $$xPw$$ or $$wPz$$ for each $$w$$ of $$X$$. The paper discusses interval orders and semiorders from the viewpoint of tolerance relations on lattices. By concentrating on properties of the associated indifference relations, it is possible to characterize interval orders as meet-tolerances and semiorders as lattice-tolerances on a chain. Some considerations are addressed to partial interval orders and semiorders, and they are related to certain tolerances on a poset.
Reviewer: I.Chajda (Přerov)

##### MSC:
 06A06 Partial orders, general 06B10 Lattice ideals, congruence relations
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