孫濤, 鄒志偉

Lawson-Mislove問(wèn)題的一個(gè)必要條件

孫濤1, 鄒志偉2

(1. 湖南文理學(xué)院 數(shù)理學(xué)院, 湖南 常德, 415000; 2. 南華大學(xué) 數(shù)理學(xué)院, 湖南 衡陽(yáng), 421001)

借助于Dcpo上的Scott拓?fù)? 引進(jìn)Scott吸收Dcpo的概念, 并證明了函數(shù)空間上Scott拓?fù)渑cIsbell拓?fù)湟恢碌谋匾獥l件是該函數(shù)空間的值域Dcpo是Scott吸收的。結(jié)果表明, Scott吸收性是Lawson-Mislove問(wèn)題的一個(gè)必要性刻畫(huà)。

函數(shù)空間; Isbell 拓?fù)? Scott 拓?fù)? Scott吸收Dcpo

1 基本概念

2 主要結(jié)果

注2 Scott吸收Dcpo的例子很多, 如完備格以及具有最大元的Dcpo等。非Scott吸收Dcpo的例子也容易得到, 如文獻(xiàn)[3]中Example3.1與Example3.2所例舉的Dcpo。

證明 采用反證法。

3 結(jié)論

本文在分析了函數(shù)空間上Isbell拓?fù)渑cScott拓?fù)渲Y(jié)構(gòu)的基礎(chǔ)上證明了二者一致的必要條件是函數(shù)空間的值域Dcpo具有Scott吸收性質(zhì)。

該結(jié)果縮小了使得Isbell拓?fù)渑cScott拓?fù)湟恢碌闹涤駾cpo的尋找范圍。對(duì)進(jìn)一步解決Lawson-Mislove問(wèn)題具有一定意義。

[1] Lawson J D, Mislove M W. Problems in domain theory and topology, in: Open Problems in Topology [M]. Amsterdam: Elsevier Science Publishers, 1990.

[2] Mill J V, Reed G M. Open Problem in Topology [M]. Amsterdam: North-Holland, 1990.

[3] Liu Y M, Liang J H. Solution to two problem of J.D. Lawson and M. Mislove [J]. Topology and its Application, 1996, 69: 153–164.

[4] Xi X Y, Yang J B. Coincidence of Isbell and Scott topologies on domain function space [J]. Topology and its Application, 2014, 164: 197–206.

[5] Gierz G, Hofmann K H, Keimel K, et al. Continuous lattices and Domains [M]. Cambridge: Cambridge University, 2003.

[6] Schwarg F, Week S. Scott topology, Isbell topology and continuous convergence [J]. Lecture Notes in Pure and Applied Mathematics, 1985, 101: 251–271.

[7] Engelking R. General Topology [M]. Warszawa: Polish Scientifific Publishers, 1977.

[8] Birkhoff G. Lattice Theory [M]. Providence: American Mathematical Society, 1940.

A necessary condition for Lawson-Mislove Problem

Sun Tao1, Zou Zhiwei2

(1. College of Mathematics and Physics, Hunan University of Arts and Science, Changde 415000, China; 2. College of Mathematics and Physics, University of South China, Hengyan 421001, China)

Based on the Scott Topology on Dcpo, the concept of Scott absorbed Dcpo is introduced. And it is proved that if the Scott Topology and the Isbell Topology on a functional space are coincident, then the valued Dcpo in the functional space is Scott absorbed. This result gives a necessary condition for Lawson-Mislove Problem.

functional space; Isbell Topology; Scott Topology; Scott absorbed Dcpo

10.3969/j.issn.1672–6146.2021.01.001

O 189.11

A

1672–6146(2021)01–0001–04

孫濤, suntao5771@163.com。

2020–09–25

國(guó)家自然科學(xué)基金項(xiàng)目(11901194&11801264); 湖南省自然科學(xué)基金(2019JJ50406); 湖南省高校青年骨干教師資助項(xiàng)目。

(責(zé)任編校: 張紅)