The algebraic extension
B
Z
F
of an extended bicyclic semigroup is introduced for an arbitrary ω -closed family ℱ of subsets of ω. It is proved that
B
Z
F
is a combinatorial inverse semigroup. We ...describe Green’s relations, the natural partial ordering on the semigroup
B
Z
F
,
and its set of idempotents. We also establish the criteria of simplicity, 0-simplicity, bisimplicity, and 0-bisimplicity of the semigroup
B
Z
F
and a criterion for
B
Z
F
to be isomorphic either to the extended bicyclic semigroup or to the countable semigroup of matrix units. It is proved that, in the case where the family ℱ consists of all singletons of ω and the empty set, the semigroup
B
Z
F
is isomorphic to the Brandt λ-extension of the semilattice (ω,min).
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, SIK, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Every gauge inverse submonoid (including Jones-Lawson's gauge inverse submonoid of the polycyclic monoid $P_{n}$) is a normal submonoid. In 2018, Alyamani and Gilbert introduced an equivalence ...relation on an inverse semigroup associated to a normal inverse subsemigroup. The corresponding quotient set leads to an ordered groupoid. In this note we shall show that this ordered groupoid is inductive if the normal inverse subsemigroup is a gauge inverse submonoid and the corresponding quotient inverse semigroup by any guage inverse submonoid is isomorphic either to the bicyclic semigroup or to the bicyclic semigroup with adjoined zero.
We are interested in properties, especially
injectivity
(in the sense of category theory), of the ternary rings of operators generated by certain subsets of an inverse semigroup via the regular ...representation. We determine all subsets of the extended bicyclic semigroup which are closed under the triple product
x
y
∗
z
(called semiheaps) and show that the weakly closed ternary rings of operators generated by them are injective operator spaces.
It is shown that every Hausdorff locally compact semigroup topology on the extended bicyclic semigroup with adjoined zero
C
ℤ
0
is discrete. At the same time, on
C
ℤ
0
, there exist 𝔠 different ...Hausdorff locally compact shift-continuous topologies. In addition, on
C
ℤ
0
, we construct a unique minimal shift-continuous topology and a unique minimal inverse semigroup topology.
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, SIK, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
A topology τ on a monoid S is called shift-continuous if for every a, b ∈ S the two-sided shift S → S, x ↦ axb, is continuous. For every ordinal α ≤ ω, we describe all shift-continuous locally ...compact Hausdorff topologies on the α-bicyclic monoid B
. More precisely, we prove that the lattice of shift-continuous locally compact Hausdorff topologies on B
is anti-isomorphic to the segment of 1, α of ordinals, endowed with the natural well-order. Also we prove that for each ordinal α the α + 1-bicyclic monoid B
is isomorphic to the Bruck extension of the α-bicyclic monoid B
.
In recent years, there have been considerable interests in the study of when a closed convex subset
K of a Banach space has the fixed point property, i.e. whenever
T is a non-expansive mapping from
K ...into
K, then
K contains a fixed point for
T. In this paper we shall study fixed point properties of semigroups of non-expansive mappings on weakly compact convex subsets of a Banach space (or, more generally, a locally convex space). By considering the classes of bicyclic semigroups we answer two open questions, one posted earlier by the first author in 1976 (Dalhousie) and the other posted by T. Mitchell in 1984 (Virginia). We also provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitopological semigroups in terms of fixed point property for non-expansive mappings related to another open problem raised by the first author in 1976.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In the present paper, a complete description of the inverse subsemigroups of the bicyclic semigroup is given. Isomorphism conditions for two inverse subsemigroups are found. From the inverse ...subsemigroups under consideration, a category is constructed, and the existence of a functor from this category to the category of finite ordered sets of numbers is proved.
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
In this paper we study submonoids of the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let ...$\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_{\infty}^{\underline{1}}$ of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}^{\underline{1}}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is the minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.
We study algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup
C
(
p
,
q
)
. We prove that a topological semigroup
S with pseudocompact square ...contains no dense copy of
C
(
p
,
q
)
. On the other hand, we construct a (consistent) example of a pseudocompact (countably compact) Tychonoff semigroup containing a copy of
C
(
p
,
q
)
.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
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
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