On monoids of monotone injective partial selfmaps of Ln ×lex Z with co-finite domains and images

• Main Authors: Gutik, O., Pozdnyakova, I.
• Format: Article
• Language: English
• Published: Інститут прикладної математики і механіки НАН України 2014
• Series: Algebra and Discrete Mathematics
• Online Access: http://dspace.nbuv.gov.ua/handle/123456789/153337

We study the semigroup IO∞(Zⁿlex) of monotone injective partial selfmaps of the set of Ln × lex Z having co-finite domain and image, where Ln ×lex Z is the lexicographic product of n-elements chain and the set of integers with the usual order. We show that IO∞(Zⁿlex) is bisimple and establish its projective congruences. We prove that IO∞(Zⁿlex) is finitely generated, and for n = 1 every automorphism of IO∞(Zⁿlex) is inner and show that in the case n ⩾ 2 the semigroup IO∞(Zⁿlex) has non-inner automorphisms. Also we show that every Baire topology τ on IO∞(Znlex) such that (IO∞(Znlex),τ) is a Hausdorff semitopological semigroup is discrete, construct a non-discrete Hausdorff semigroup inverse topology on IO∞(Zⁿlex), and prove that the discrete semigroup IO∞(Zⁿlex) cannot be embedded into some classes of compact-like topological semigroups and that its remainder under the closure in a topological semigroup S is an ideal in S.