Algebraic Curves and Projective Geometry by Edoardo Ballico, Ciro Ciliberto

By Edoardo Ballico, Ciro Ciliberto

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This does not hold for every chain, even if this chain is a maximal one: In the four-element set {a, b, c, d) where a < b < c, a < d < c and b 11 d (see Figure 4) we have: {a, b, c} is a maximal chain, but it does not contain all elements between a and c because d is missing. 8 Theorem and Definition. Let a be an element of a poset (P,5) and Inc(a) the set of all elements of P which are incomparable with a. Then Inc(a) is convex. Proof. Let x, y E Inc(a) and x 5 z 5 y for an element z E P. Then a I z cannot hold, otherwise we would also have a I y contradicting y 11 a.

Contrary to this the values max T and min T are determined internally. 10 Definition. An element a of a poset (P,5 ) is said to be minimal (resp. maximal), if no element x of P exists which satisfies x < a (resp. x > a). We denote the set of all minimal (resp. maximal) elements of P by Mi(P) (resp. Ma(P)). 11 Remark. If a poset has a least element, then this is also a minimal element, analogously the greatest element is also a maximal element. In a linearly ordered set the notions "minimal element" and "least element" evidently coincide, also "maximal" and "greatest".

Is procedure is called the dualization of the first proof. g. lower and upper bound, inf and sup, min and max. The duality principle of order theory implies the duality principle of lattice theory, which (in consequence of the exchange of inf and sup) implies the exchange of A by V and conversely. This page intentionally left blank Chapter 2 General relations between posets and their chains and antichains In this chapter we treat several structure theorems for posets, for which no special assumptions are required.

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