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Description of the Orders Which Are Hereditarily Half-reconstructible by Their Comparability Graphs
Mohammad Alzohairi, Moncef Bouaziz, Youssef Boudabbous and Ahmad Sharary

Let P and P’ be two orders on the same set X. The order P’ is hemimorphic to P if it isomorphic to P or to its dual P. It is hereditarily hemimorphic to P if for each subset A of X, the orders P⨡A and P’⨡A induced on A are hemimorphic. The order P is hereditarily half-reconstructible by its comparability graph if it is hereditarily hemimorphic to each order Q on X having the same comparability graph as P.

In this paper, we begin by obtaining a new result on the decomposition of orders. Then we use this result to describe the orders which are hereditarily half-reconstructible by their comparability graphs. By the last result we solve an open problem posed in 2013, by Alzohairi, Bouaziz, and Boudabbous.

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