THE PT-ORDER AND THE FIXED-POINT PROPERTY

The PT-order, or passing through order, of a poset P is a quasi order left-pointing triangle with bar underneath defined on P so that a left-pointing triangle with bar underneath b holds if and only if every maximal chain of P which passes through 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 left-pointing triangle with bar underneath-maximal, (ii) X is a left-pointing triangle with bar underneath-antichain, and (iii) X is left-pointing triangle with bar underneath-dominating; we call such a subset a left-pointing triangle with bar underneath-good subset of P. A left-pointing triangle with bar underneath-good subset is a retract of P and any two left-pointing triangle with bar underneath-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 left-pointing triangle with bar underneath-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 left-pointing triangle with bar underneath-good subset of itself) which is the transfinite analogue of the core of a finite poset obtained by dismantling.