Proceedings of the International Geometry Center

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On the monoid of cofinite partial isometries of $\qq{N}^n$ with the usual metric

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Published Dec 28, 2019
DOI https://doi.org/10.15673/tmgc.v12i3.1553

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Oleg Gutik
Ivan Franko National University of Lviv
http://orcid.org/0000-0001-8513-0282
Anatolii Savchuk
Ivan Franko National University of Lviv

Abstract

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.

Keywords:
Partial isometry, inverse semigroup, partial bijection, natural partial order, Green's relations, least group congruence, F-inverse semigroup, semidirect product, free semilattice, symmetric group

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How to Cite
Gutik, O., & Savchuk, A. (2019). On the monoid of cofinite partial isometries of $\qq{N}^n$ with the usual metric. Proceedings of the International Geometry Center, 12(3), 51-68. https://doi.org/10.15673/tmgc.v12i3.1553
Issue
Vol 12 No 3 (2019)
Section
Papers

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Author Biographies

Oleg Gutik, Ivan Franko National University of Lviv

E.g., department and rank Deprtment of Geometry and Topology. Assoc/ Prof.

Anatolii Savchuk, Ivan Franko National University of Lviv

Dept. of Algebra and Logic

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