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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 chain
However, 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).
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