Fixpoint Theorem for Continuous Functions on Chain-Complete Posets

Fixpoint Theorem for Continuous Functions on Chain-Complete Posets

Let P be a non empty poset. Observe that there exists a chain of P which is non empty. Let I1 be a relational structure. We say that I1 is chain-complete if and only if: (Def. 1) I1 is lower-bounded and for every chain L of I1 such that L is non empty holds sup L exists in I1. One can prove the following proposition (1) Let P1, P2 be non empty posets, K be a non empty chain of P1, and h be a mo...

Meet-continuity on $L$-directed Complete Posets

In this paper, the definition of meet-continuity on $L$-directedcomplete posets (for short, $L$-dcpos) is introduced. As ageneralization of meet-continuity on crisp dcpos, meet-continuity on$L$-dcpos, based on the generalized Scott topology, ischaracterized. In particular, it is shown that every continuous$L$-dcpo is meet-continuous and $L$-continuous retracts ofmeet-continuous $L$-dcpos are al...

On Quillen's Theorem A for posets

A theorem of McCord of 1966 and Quillen’s Theorem A of 1973 provide sufficient conditions for a map between two posets to be a homotopy equivalence at the level of complexes. We give an alternative elementary proof of this result and we deduce also a stronger statement: under the hypotheses of the theorem, the map is not only a homotopy equivalence but a simple homotopy equivalence. This leads ...

A Theorem on Semi-Continuous Functions

A Theorem on Semi - Continuous Functions

RECENTLY G. C. Young* and A. Denjoyf have communicated theorems—those in Denjoy's memoir are of an especially comprehensive character—dealing, in particular, with point sets where the four derivatives of a given continuous function are identical. It is the purpose of this note to treat an analogous problem that arises when "derivative" is replaced by "saltus."$ However, instead of confining our...