Open AccessOn the monoid of cofinite partial isometries of $\qq{N}^n$ with the usual metric
Author(s) -
Олег Гутік,
Anatolii Savchuk
Publication year - 2019
Publication title -
trudy meždunarodnogo geometričeskogo centra/pracì mìžnarodnogo geometričnogo centru
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.124
H-Index - 2
eISSN - 2409-8906
pISSN - 2072-9812
DOI - 10.15673/tmgc.v12i3.1553
Subject(s) - mathematics , semidirect product , monoid , semilattice , semigroup , combinatorics , quotient , congruence (geometry) , group (periodic table) , bicyclic semigroup , integer (computer science) , cancellative semigroup , unit (ring theory) , product (mathematics) , pure mathematics , chemistry , geometry , mathematics education , organic chemistry , computer science , programming language
In this paper we study the structure of the monoid Iℕn ∞ of cofinite partial isometries of the n-th power of the set of positive integers ℕ with the usual metric for a positive integer n > 2. We describe the group of units and the subset of idempotents of the semigroup Iℕn ∞, the natural partial order and Green's relations on Iℕn ∞. In particular we show that the quotient semigroup Iℕn ∞/Cmg, where Cmg is the minimum group congruence on Iℕn ∞, is isomorphic to the symmetric group Sn and D = J in Iℕn ∞. Also, we prove that for any integer n ≥2 the semigroup Iℕn ∞ is isomorphic to the semidirect product Sn ×h(P∞(Nn); U) of the free semilattice with the unit (P∞(Nn); U) by the symmetric group Sn.