Can I read The Relation Between the Height of a Well-founded Partial Ordering and the Order Types of Its Chains and Antichains on EtoBox?

The Relation Between the Height of a Well-founded Partial Ordering and the Order Types of Its Chains and Antichains by Diana Schmidt is a Computer Science article available to read on EtoBox.

What is The Relation Between the Height of a Well-founded Partial Ordering and the Order Types of Its Chains and Antichains about?

We show that a well-founded partial ordering of countable height a must contain either a chain of order type a or an antichain of order type ~o. On the other hand, there are partial orderings of arbitrary countable height with no infinite chains and no antichains of order type (co + 1). DEFINITIONS. Any well-founded partial ordering < on a set X associates an ordinal with each element of X in a natural way: For x E X, the height of x (with respect to (X, <)) is defined by The height of the partial ordering (X, <) can then be defined by ht(X, <) = sup{ht(x) + 1 ] x E X}. ht(X, <) is the least ordinal into which (X, <) can be embedded in an orderpreserving way. A subset Y of X is a chain (with respect to (X, <)) if it is linearly ordered by <, an antichain (with respect to (X, <)) if no element of Y • Y belongs to <. The order type of Y (with respect to (X, <)) is the order type of the set tht(x) tx ~ Y} of ordinals with respect to the usual ordering on the ordinals. Note that the order type of Y depends on the whole of <, not just on its restriction to Y; for example, the order type of Y is equal to ht(Y, </" Y) if Y is a chain, but if Y is an antichain ht(Y, </'Y)= 1, whereas the o

Who reads The Relation Between the Height of a Well-founded Partial Ordering and the Order Types of Its Chains and Antichains?

Author

Diana Schmidt

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0095-8956)

Published

1981

Language

EN

Field

Computer Science (Physical Sciences)