2005/11/15 by Ganna Kudryavtseva, Volodymyr Mazorchuk · 1 citation
Mathematics · #math.GR #msc:20M10 #msc:20M05
published as Yokohama Math. J. 55 (2009), no. 1, 21-46 · 26 pages
arxiv created 2005/11/15 · arxiv updated 2010/04/02
We study the algebraic properties of the series Kn of semigroups, which is inspired by \citeKi and has origins in convexity theory. In particular, we describe Green's relations on Kn, prove that there exists a faithful representation of Kn by n× n matrices with non-negative integer coefficients (and even explicitly construct such a representation), and prove that Kn does not admit a faithful representation by matrices of smaller size. We also describe the maximal nilpotent subsemigroups in Kn, all isolated and completely isolated subsemigroups, all automorphisms and anti-automorphisms of Kn. Finally, we explicitly construct all irreducible representations of Kn over any field and describe primitive idempotents in the semigroup algebra (which we prove is basic).