Continuous Lattices: Proceedings of the Conference on by Bernhard Banaschewski (auth.), Bernhard Banaschewski,

By Bernhard Banaschewski (auth.), Bernhard Banaschewski, Rudolf-Eberhard Hoffmann (eds.)

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Additional info for Continuous Lattices: Proceedings of the Conference on Topological and Categorical Aspects of Continuous Lattices (Workshop IV) Held at the University of Bremen, Germany, November 9–11, 1979

Example text

Precontinuous this lattices, of of the is c o m p l e t e l y but the w a y - b e l o w relation 47 2. P r e c o n t i n u o u s posets Let P be a partially is d e n o t e d ~ . We adopt +y : = { x principal is t h e is t h e lower @(P) ~J { +y : arbitrary sets The map Y ~ the is c a l l e d = lattice p : P complete By a for all (Prink) finite is an ideal, ideals bitrary (see (algebraic) [Eli). sets sets form are a topology open. lower bounds by Y o f Y, > dually. Y+and respectively. y ~ x } " Thus C is a c u t called the iff C = C a.

Erreichen wir v 2 _ vIT v O l I I uo uI u2 andererseits die k H r z e r e =V ! i := I ..... n-1 i U =U n (Po'qi) Folge (Uo,V) ' (po,q~) = E(Po,q*). : 30 3. Beweis des V e r f e i n e r u n g s s a t z e s Es s e i e n jetzt erfOllt. Wir U×V geordneten des haben die Voraussetzungen insbesondere hiermit Wir w~hlen ein sei Ferner haben wir aus §2 allemal . Ferner einen setzen Punkt wit das (po,qo) E P×Q = Pru(E(pX{qo})) P(V) = p r p ( E -I ({Uo}×V)) , V(P) = Prv(E(P×{qo})) Q(U) = p r Q ( E -I (U×{Vo})) , U(Q) = Pru(E({Po}×Q)) Q(V) = p r Q ( E -I ({Uo}×V)) , V(Q) = Prv(E({Po}XQ)) als ein und setzen : U(P) erstes .

D i s k r e t e n Fall ist klar, was unter O r d n u n g s z u s a m m e n h a n g ist: Je zwei E l e m e n t e m e n t e n verbinden, Im zu v e r s t e h e n lassen sich dutch eine e n d l i c h e Folge von Ele- in der a u f e i n a n d e r f o l g e n d e E l e m e n t e v e r g l e i c h b a r sind. Mit diesme rein o r d n u n g s t h e o r e t i s c h e n Zusammenhangsbegriff ist die V e r a l l g e m e i n e r u n g des Satzes von H a s h i m o t o auf den t o p o l o g i s c h e n Fall nicht schwierig.

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