Continuous Homomorphic Images of Real Clans with Zero
Haskell Cohen, I. S. Krule · Proceedings of the American Mathematical Society · 1959
A clan is a compact connected topological semigroup with identity element.We call a clan a real clan provided its underlying topological space is a closed interval of real numbers with the usual topology.Recently A. D. Wallace conjectured that the continuous homomorphic image of a real clan with zero is again one such.We show that under suitable conditions this conjecture is true, but that in general the image is either a real clan with zero or a triad (i.e., homeomorphic to the letter "T" without serifs); moreover, if the image is a triad, then its zero is an endpoint.Throughout this paper 5 denotes a real clan with zero, u its identity, z its zero, and E its set of idempotents, and h is a continuous homomorphism of S onto the clan T.