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

• Main Authors: Oleg Gutik, Anatolii Savchuk
• Format: Article
• Language: Russian
• Published: Odessa National Academy of Food Technologies 2019-12-01
• Series: Pracì Mìžnarodnogo Geometričnogo Centru
• Subjects: partial isometry, inverse semigroup, partial bijection, natural partial order, green's relations, least group congruence, f-inverse semigroup, semidirect product, free semilattice, symmetric group
• Online Access: https://journals.onaft.edu.ua/index.php/geometry/article/view/1553

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.