A Type of Algebraic Structure Related to Sets of Intervals

Abstract

F. Wehrung has asked: Given a family \(\mathcal {C}\) of subsets of a set Ω, under what conditions will there exist a total ordering on Ω with respect to which every member of \(\mathcal {C}\) is convex?¶ We look at the family \(\mathcal {P}\) of subsets of Ω generated by \(\mathcal {C}\) under certain partial operations which, when Ω is given with a total ordering, preserve convexity; we determine the possible structures of \(\mathcal {P}\) under these operations if \(\mathcal {C},\) and hence \(\mathcal {P},\) is finite, and note a condition on that structure that is necessary and sufficient for there to exist an ordering of Ω of the desired sort. From this we obtain a criterion which works without the finiteness hypothesis on \(\mathcal {C}\).¶ Bounds are obtained on the cardinality of the set \(\mathcal {P}\) generated under these operations by an n-element set \(\mathcal {C}\).¶ We end by noting some other ways of answering Wehrung’s question, using results in the literature. The bibliography lists still more related literature.