Advanced algebra by Rotman J.J.

By Rotman J.J.

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This order is called reverse, or dual, order. The ordered set (A, >*) is called the ordered set dual to A and is denoted by A *. Thus, the identity mapping 1A : A -+ A is a duality A -+ A *. The very simple fact of the existence of A * for each ordered set A has useful consequences which seem to belie the triviality of this fact. The most important of these is the duality principle which asserts that for every theorem about ordered sets there is a dual theorem obtained simply by reversing order.

2 Let 71, as usual, denote the set of integers. If a, b E 71, write a b or b > a. In the former case a = a V b and b = a /I. b. 4 Any "geometric" lattice is a lattice. By geometric lattice we mean a configuration, oriented as in the diagram, consisting of two classes of parallel lines that intersect each other.

If A and B are ordered sets, then a mapping I: A -'J>- B is an order homomorphism provided that a>b§t(a»t(b). 'if a, b EA. Then t is an order injection (resp. surjection, bijection) in case t is an order homomorphism and an injective (resp. surjective, bijective) mapping. An order isomorphism I: A -'J>- B is a bijection such that both I and 1-1 are order homomorphisms. Note that every order injection I: A ->- B induces an order isomorphism I: A -'J>- im t, hence, any bijective order homomorphism is an order isomorphism.

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