Projects per year
Abstract
The relative rank rank(S : U) of a subsemigroup U of a semigroup S is the minimum size of a set V subset of S such that U together with V generates the whole of S. As a consequence of a result of Sierpinski, it follows that for U <= TX, the monoid of all selfmaps of an infinite set X, rank(TX : U) is either 0, 1 or 2, or uncountable. In this paper, the relative ranks rank(TX : Ox) are considered, where X is a countably infinite partially ordered set and Ox is the endomorphism monoid of X. We show that rank(TX : OX) <= 2 if and only if either: there exists at least one element in X which is greater than, or less than, an infinite number of elements of X; or X has vertical bar X vertical bar connected components. Four examples are given of posets where the minimum number of members of TX that need to be adjoined to OX to form a generating set is, respectively, 0, 1, 2 and uncountable.
Original language  English 

Pages (fromto)  177191 
Number of pages  15 
Journal  Bulletin of the London Mathematical Society 
Volume  38 
Issue number  2 
DOIs  
Publication status  Published  Apr 2006 
Keywords
 TRANSFORMATION SEMIGROUPS
Fingerprint
Dive into the research topics of 'Rank properties of endomorphisms of infinite partially ordered sets'. Together they form a unique fingerprint.Projects
 1 Finished

EP/C523229/1: Multidisciplinary Critical Mass in Computational Algebra and Applications
Linton, S. A., Gent, I. P., Leonhardt, U., Mackenzie, A., Miguel, I. J., Quick, M. & Ruskuc, N.
1/09/05 → 31/08/10
Project: Standard