Linear Extension Majority Cycles for Small (n <= 9) Partial Orders.

Let P denote a partial order on a set X of cardinality n. A linear order L is a linear extension of P if P contains or is equal to L. scriptL(P) denotes the set of all linear extensions of P and, for all x,y elements of X, scriptL(x,y) is the subset of scriptL(P) which has L element of scriptL(x,y) if xLy. A linear extension majority relation M on X is defined by xMy if #scriptL(x,y) > #scriptL(y,x); a LEM cycle exists if there are x,y,z element of X with xMyMzMx. With M' on X defined by xM'y if x does not= y and #scriptL(x,y) >= # scriptL(y,x) a LEM quasi-cycle exists if there are x,y,z element of X with xM'yM'zM'x and the equality part of M' holds for exactly one pair from (x,y,z). The study shows that no LEM cycles or LEM quasi-cycles exist on X for n