Note that the

maximal chain from 0 to a maximal element of J maybe not unique, and the expression [[union].sup.r.sub.i=l][[bar.L].sub.i] may be not unique in general.

If there exists a

maximal chain consisting of left-modular elements, then P is called left-modular.

The rank rk(x) is the length r of a

maximal chain [C.sup.m] < [X.sub.1] < xxx < [x.sub.r] = x of maximal length.

Moreover, by the standard chain decomposition of network flows of Ford Jr and Fulkerson (2010) (essentially Stanley's transfer map), which expresses g as a sum of positive flows through each

maximal chain of P, it is clear that for A an antichain of P, we have that e(P) [greater than or equal to] [[summation].sub.[upsilon] [member of] A] [[summation].sub.(x,[upsilon]) [member of] [??]] g(x, [upsilon]), since antichains intersect

maximal chains of P at most once.

Let (P, [[less than or equal to].sub.P]) be a bounded poset and let c : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a

maximal chain of P.

In particular, we say that a

maximal chain C in SP(u, v) is rising if the path corresponding to C in B(u, v) is rising.

In this situation, we say that C is a

maximal chain for e.

To the sequence ([c.sub.1], [c.sub.2], ..., [c.sub.r]) corresponds a unique

maximal chainHowever, we do not use all such

maximal chains since otherwise Grassmannians of infinite rank would in general give rise to a disconnected chamber system, which is of course not desirable.

Recall that a finite poset is called graded, if all

maximal chains are of the same length.

Let M (P) denote the set of

maximal chains of P and let M'(P) denote the set of chains of length l - 1.

A Sheffer poset is a graded poset such that the number of

maximal chains D(n) in an n-interval [[??], y] depends only on [rho](y) = n, the rank of the element y, and the number B(n) of

maximal chains in an n-interval [x, y], where x [not equal to] [??], depends only on [rho](x, y) = [rho](y) - [rho](x).