Orders: Description and Roles by Maurice Pouzet

By Maurice Pouzet

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A BA . well-founded) is n o t c o u n t a b l y g e n e r a t e d , s o I A function and = sup{ [ T I : T An a l g e b r a . h a s a c o f i n a l s u b s e t of power A The a l g e b r a c o n s t r u c t e d by S h e l a h [ 8 1 ] assumi ng h-cofinality with A ? A of but has by R a u m g a r t n e r , Komjath 1 5 (d} h-c of(A ) with A w1 = I i s a maximal = w1 . < incA h-c of(A ) ? Algebraic Density . nA = min{lDI : D d e n s e i n A} Again t h i s is a l a r g e f u n c t i o n . T h i s is t h e same a s t h e n-weight of u l t A , but we s h a l l u s u a l l y c a l l i t a l g e b r a i c I t h a s been e x t e n s i v e l y s t u d i e d in t o p o l o g y .

A. T o Professor E . s sur les algsbres booleennes. Nous btudions par exemple les applications : B est une image homomorphe infinie de A } , la longueur h A = min { I l l length A = s u p { I X I : X est une chaine de A} , et le nombre d e s ultrafiltres de A , Ult A Nous d6crivons complhtement en details 24 telles applications, et nous en mentionnons 16 autres. Nous pensons que l'6tude de c e s applications cardinales fournit un cadre commode pour analyser la structure des algkbres boolbennes et que c'est un bon terrain d'expbrimentation pour les mbthodes recentes de demonstrtltion en thPorie des ensembles.

In p a r t i c u l a r , t h e following q u e s t i o n s a r e open. P r o b l e m 20. Is l S u b A \ a l w a y s a power of Problem 21. W A l L Can o n e p r o v e i n w ? For \A\ = \SubA\ < ZK ? R+>A , then If and A s; R or t h a t t h e r e is a ZFC A RA with s i n g u l a r is i t c o n s i s t e n t t h a t t h e r e i s a Problem 2 Z S . K 2 ? K \5 \SubA 8. lSubR BA A with I . Irredundance . A s u b s e t X o f A i s i r r e d u n d a n t i f Vx E X ( x f S g ( X \ {XI) We l e t irrA = sup{\X\ : X irredundant} T h i s is a l a r g e f u n c t i o n .

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