On the monoid of cofinite partial isometries of N with the usual metric

Abstract: In the paper we show that the monoid

m a t h b f I m a t h b b N_{i n f t y}

of all partial cofinite isometries of positive integers does not embed isomorphically into the monoid

m a t h b f {I D}_{i n f t y}

of all partial cofinite isometries of integers. Moreover, every non-annihilating homomorphism

m a t h f r a k h c o l o n m a t h b f I m a t h b b N_{i n f t y} o m a t h b f {I D}_{i n f t y}

has the following property: the image

(m a t h b f I m a t h b b N_{i n f t y}) m a t h f r a k h

is isomorphic either to the two-element cyclic group

m a t h b b Z_{2}

or to the additive group of integers

m a t h b b Z (+)

. Also we prove that the monoid

m a t h b f I m a t h b b N_{i n f t y}

is not finitely generated, and, moreover, monoid

m a t h b f I m a t h b b N_{i n f t y}

does not contain a minimal generating set.