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