We study the semigroup I
of injective partial cofinite selfmaps of an infinite cardinal λ. We show that I
is a bisimple inverse semigroup and each chain of idempotents in I
is contained in a bicyclic ...subsemigroup of I
, we describe the Green relations on I
and we prove that every non-trivial congruence on I
is a group congruence. Also, we describe the structure of the quotient semigroup I
/σ, where σ is the least group congruence on I
We study the semigroup
of monotone
injective partial selfmaps of the set of integers having cofinite
domain and image. We show that
is bisimple and all
of its non-trivial semigroup homomorphisms are ...either isomorphisms
or group homomorphisms. We also prove that every Baire topology
on
, such that
is a
Hausdorff semitopological semigroup, is discrete and we construct a
non-discrete Hausdorff inverse semigroup topology
on
. We show that the
discrete semigroup
cannot be embedded into some classes of compact-like topological
semigroups and that its remainder under the closure in a topological
semigroup
is an ideal in
We study the semigroup of monotone injective partial selfmaps of the set of integers having cofinite domain and image. We show that is bisimple and all of its non-trivial semigroup homomorphisms are ...either isomorphisms or group homomorphisms. We also prove that every Baire topology on , such that is a Hausdorff semitopological semigroup, is discrete and we construct a non-discrete Hausdorff inverse semigroup topology on . We show that the discrete semigroup cannot be embedded into some classes of compact-like topological semigroups and that its remainder under the closure in a topological semigroup S is an ideal in S.
In this paper we study the semigroup
of injective partial selfmaps almost everywhere the identity of a set of infinite cardinality
. We describe the Green relations on
, all (two-sided) ideals and ...all congruences of the semigroup
. We prove that every Hausdorff hereditary Baire topology
on
such that (
) is a semitopological semigroup is discrete and describe the closure of the discrete semigroup
in a topological semigroup. Also we show that for an infinite cardinal
the discrete semigroup
does not embed into a compact topological semigroup and construct two non-discrete Hausdorff topologies turning
into a topological inverse semigroup.
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