32,563 research outputs found
Linear Extensions and Comparable Pairs in Partial Orders
We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also show that a random interval partial order on elements has close to a third of the pairs comparable with high probability, and the number of linear extensions is with high probability
Linear extensions of partial orders and Reverse Mathematics
We introduce the notion of \tau-like partial order, where \tau is one of the
linear order types \omega, \omega*, \omega+\omega*, and \zeta. For example,
being \omega-like means that every element has finitely many predecessors,
while being \zeta-like means that every interval is finite. We consider
statements of the form "any \tau-like partial order has a \tau-like linear
extension" and "any \tau-like partial order is embeddable into \tau" (when
\tau\ is \zeta\ this result appears to be new). Working in the framework of
reverse mathematics, we show that these statements are equivalent either to
B\Sigma^0_2 or to ACA_0 over the usual base system RCA_0.Comment: 8 pages, minor changes suggested by referee. To appear in MLQ -
Mathematical Logic Quarterl
- …