**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. 69, 2 (2000)

pp. 241-259

CLONES, COCLONES AND COCONNECTED SPACES

V. TRNKOVA

**Abstract**.
Clones and coclones motivate this examination of coconnected spaces. A space $X$ is coconnected if every continuous map $X\times X\to X$ depends only on one variable. We prove here that every monoid can be represented as the monoid of all nonconstant continuous selfmaps of a coconnected space and that, within the class of Hausdorff spaces, the coconnectedness is not expressible by a sentence of the first order language of the monoid theory: we construct two Hausdorff spaces with isomorphic monoids of all continuous selfmaps such that one of them is coconnected and the other is not.

**AMS subject classification**.
54C05, 08A10

**Keywords**.
Clone, the first order language of clone theory, dual notions, the first order language of monoid theory, connected topological space, monoid of continuous selfmaps of a space, continuous binary operation

**Download:** Adobe PDF Compressed Postscript

Acta Mathematica Universitatis Comenianae

Institute of Applied
Mathematics

Faculty of Mathematics,
Physics and Informatics

Comenius University

842 48 Bratislava, Slovak Republic

Telephone: + 421-2-60295111 Fax: + 421-2-65425882

e-Mail: amuc@fmph.uniba.sk
Internet: www.iam.fmph.uniba.sk/amuc
© Copyright 2001, ACTA MATHEMATICA
UNIVERSITATIS COMENIANAE