ISSN:
1572-9273
Keywords:
06A06
;
(Partially) ordered set
;
PT-order
;
chain complete
;
retract
;
fixed-point property
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract The PT-order, or passing through order, of a poset P is a quasi order ⊴ defined on P so that a⊴b holds if and only if every maximal chain of P which passes throug a also passes through b. We show that if P is chain complete, then it contains a subset X which has the properties that (i) each element of X is ⊴-maximal, (ii) X is a ⊴-antichain, and (iii) X is ⊴-dominating; we call such a subset a ⊴-good subset of P. A ⊴-good subset is a retract of P and any two ⊴-good subsets are order isomorphic. It is also shown that if P is chain complete, then it has the fixed point property if and only if a ⊴-good subset also has the fixed point property. Since a retract of a chain complete poset is also chain complete, the construction may be iterated transfinitely. This leads to the notion of the “core” of P (a ⊴-good subset of itself) which is the transfinite analogue of the core of a finite poset obtained by dismantling.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00420351
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