On Partially Ordered Sets with Small Initial Segments

Aleksander Rutkowski
1985 Demonstratio Mathematica  
Dedicated to the memory o f Pro fessor Roman Sikorski I" Introduction Wa base our considerations on Zermelo-Fraenkel sat theory with the axiom of choice. In particular: an ordinal coincides with the set of all its predecessors, a cardinal is an initial ordinal. Every well ordered set is isomorphic with an ordinal (ordered by e) and every set X oan be well ordered isomorphi r cally to a oertain cardinal {which is called the cardinality of X and denoted by |X|). ' We denote cartesian produot of
more » ... tesian produot of the family {A i :ie i} by IT A< and its cardinality by fT . If (Vie I) |A<| -m ie I x iel 1 1 • and Hi = n, then ^TT^!A^f is denoted by m 11 « It is easy to see that Lot ^ be a partial order of a se.t P and x,y e P. We call x 8 y incomparable if x*£y and y^x" A set of mutually incomparable elements is called antiohain. An m-antichain is an antichain of cardinality m e Every antichain is included in a uaximal antiohain. -177 -Unauthenticated Download Date | 2/25/20 5:08 PM
doi:10.1515/dema-1985-0115 fatcat:szsv34bilfauvcba33fzkxbwyi