On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers
In this paper we study submonoids of the monoid I_∞^ \!\! earrow(N) of almost monotone injective co-finite partial selfmaps of positive integers N. Let I_∞^ earrow(N) be a submonoid of I_∞^ \!\! earrow(N) which consists of cofinite monotone partial bijections of N and C_N be a subsemigroup of I_∞^ \!\! earrow(N) which is generated by the partial shift nn+1 and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of I_∞^\! earrow(N) which contains the semigroup C_N is the identity map. We construct a submonoid IN_∞^[1] of I_∞^ \!\! earrow(N) with the following property: if S is an inverse submonoid of I_∞^ \!\! earrow(N) such that S contains IN_∞^[1] as a submonoid, then every non-identity congruence C on S is a group congruence. We show that if S is an inverse submonoid of I_∞^ \!\! earrow(N) such that S contains C_N as a submonoid then S is simple and the quotient semigroup S/C_mg , where C_mg is the minimum group congruence on S, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of I_∞^ \!\! earrow(N) which contain C_N and embeddings of such semigroups into compact-like topological semigroups. , number=2, journal=Carpathian Mathematical Publications, author=Gutik, O.V. and Savchuk, A.S., year=2019, month=Dec., pages=296–310