On the family of linear extensions of a partial order

AbstractGiven a partial order P defined on a finite set X, a binary relation ≻P may be defined on X by setting x ≻P y 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 partial orders P such that ≻P includes cycles. Thus, in a voting situation in which voters are unanimous in their preferences on the pairs in P and express all possible linearly ordered preferences on X which are consistent with P, with no two voters having the same preference order, strict simple majorities as given by ≻P can cycle