Linear Extension Majority Cycles on Partial Orders.

Let P define a partial order on a set X of cardinality n. A linear extension L of P is a linear order with P $c under L, and $font M2 L(P) is the set of all linear extensions of P. $font M2 L(x,y) denotes that subset of $font M2 L(P) with xLy for x,y epsilon X. A linear extension majority (LEM) relation M on X is defined by xMy xM'y if #$font M2 L(x,y) >= #$font M2 L(y,x). A LEM cycle exists if there are x,y,z epsilon X with xMyMzMx, and a LEM quasi-cycle exists if xM'yM'zM'x and the equality part of the definition of M' holds for exactly one pair in the triple.