By Sik Kim Hee, Joseph Neggers

An advent to the idea of partially-ordered units, or "posets". The textual content is gifted in really an off-the-cuff demeanour, with examples and computations, which depend upon the Hasse diagram to construct graphical instinct for the constitution of limitless posets. The proofs of a small variety of theorems is integrated within the appendix. vital examples, in particular the Letter N poset, which performs a task reminiscent of that of the Petersen graph in offering a candidate counterexample to many propositions, are used time and again through the textual content.

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42 Chapter 3 Poset Morphisms As in other structures, algebraic and otherwise, morphisms play a very important role in investigating the structure of posets. In the literature several authors use different terminologies and notations for poset morphisms. Let (X,::;) and (Y, ::;) be two partially ordered sets. A mapping f from X to Y is called order preserving if for every two elements x and y of X, x ::; y implies ple, if X= {a,b,c} with Hasse diagram with Hasse diagram I~' then f f (x) f (y). b andY= {1,2} = {(a,2),(b,2),(c,1)} is an order preserving mapping from X to Y.

A poset (X,~) is a circle order if there is a function F assigning to each x E X a circle Fx on the plane so that x ~ y in (X,~) Fx s;:; Fy· ¢:? x~y These diagrams look like Venn diagrams for representing sets, but the different points x and y are incomparable unless Fx n Fy is Fx or Fy. , a poset which cannot be represented as a circle order. Angular regions as well as circles are examples of convex region in the plane. This suggests that one may also study convex order, defined in the obvious way.

3) or /(1) = /(2) I /(2) = /(3) /(1) 2 = 6 order preserving mappings from c3 toN whose images are C 2 . Thus, there are 4 + 6 = 10 order preserving mappings from C3 toN. Since the image of a chain under order preserving mapping is also a chain, we summarize this observation as it relates to isomorphisms: 47 Let f be a one-to-one order preserving mapping from a chain X onto a poset Y. Then f is an isomorphism and Y is also a chain. ( *) Consider a finite chain. It is easy to see that except for the identity mapping there is no isomorphism from a finite chain Cn onto itself.