On linear extension majority graphs of partial orders

The linear extension majority (LEM) graph (X, > p) of a finite partially ordered set (X, P) has x>py for elements x and y in X just when more linear extensions L of P on X have xLy than yLx. A linear extension L of P on X is a linear order on X with P ? L. There exist finite partially ordered sets (X, P) whose LEM graphs have no >p-maximal elements, in which case every x in X has an x′ in X for which x′>px.