84 research outputs found
Embedding the bicyclic semigroup into countably compact topological semigroups
We study algebraic and topological properties of topological semigroups
containing a copy of the bicyclic semigroup C(p,q). We prove that each
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) Tychonov semigroup containing a copy of
C(p,q).Comment: 14 page
Presentations of factorizable inverse monoids
It is well-known that an inverse monoid is factorizable if and only if it is a homomorphic
image of a semidirect product of a semilattice (with identity) by a group.
We use this structure to describe a presentation of an arbitrary factorizable inverse
monoid in terms of presentations of its group of units and semilattice of idempotents,
together with some other data. We apply this theory to quickly deduce a well known
presentation of the symmetric inverse monoid on a nite set
Algebraosztályok és klónok = Classes of algebras and clones
Főbb eredményeink a következők. Beláttuk, hogy többségi kifejezésfüggvény létezése eldönthető véges algebrákra. Igazoltuk, hogy bármely nem-triviális idempotens Malcev-feltételt teljesítő véges algebrának van gyenge többségi kifejezésfüggvénye, és 2-uniform kongruenciái felcserélhetőek. Bizonyítottuk, hogy a k-parallelogramma-kifejezéssel rendelkező véges, reziduálisan kicsi algebrák és a véges 2-nilpotens csoportok kifejezésfüggvényeinek klónját véges sok reláció meghatározza. Új dichotómiatételeket kaptunk a kényszer-kielégíthetőségi problémára és az egyenletrendszer-megoldhatósági problémára. Számos új eredményt kaptunk hálók kombinatorikai vonatkozásairól, fraktál- és féligmoduláris hálókról. Struktúratételeket bizonyítottunk az E-tömör lokálisan inverz félcsoportokra, az E-unitér majdnem faktorizálható ortodox félcsoportokra, valamint kiterjesztettük a majdnem faktorizálható inverz félcsoportok elméletét a lokálisan inverz félcsoportok osztályára. Jellemeztünk bizonyos transzformáció-monoidokat, amelyek egyelemű monadikus intervallumot határoznak meg a klónhálóban. Leírtuk a centralizátorklónt véges, egyszerű, idempotens algebrák és bizonyos kongruencia disztributív varietást generáló véges algebrák esetén. Új eredményeket kaptunk 3-változós többségi függvénnyel rendelkező minimális klónokra. Beláttuk, hogy a kompozícióra zárt függvényosztályok hálója kontinuum számosságú a kételmű halmazon, és leírtuk e háló szerkezetét. | Our main results are as follows. We proved that the existence of a near-unanimity term operation is decidable for finite algebras. We showed that if a finite algebra admits a nontrivial idempotent Maltsev condition, then it has a weak near-unanimity term operation, and its 2-uniform congruences permute. We proved that the clone of any finite residually small algebra with a k-parallelogram term operation and any finite 2-nilpotent group is determined by finitely many relations. We obtained new dichotomy theorems for the constraint satisfaction problem and for the solvability problem of systems of equations. We proved a number of theorems on the combinatorial aspects of lattices and on fractal and semimodular lattices. We obtained new structure theorems for E-solid locally inverse semigroups and E-unitary almost factorizable orthodox semigroups. Furthermore we extended the theory of almost factorizable inverse semigroups to the class of locally inverse semigroups. We characterized certain transformation monoids which determine a one-element monoidal interval in the lattice of clones. We described the centralizer clones of finite simple idempotent algebras and of certain algebras in congruence distributive varieties. We obtained new results on the minimal clones containing majority operations. We proved that on the two-element set the lattice of function classes closed under composition has the cardinality of the continuum, and described the structure of this lattice
On a semitopological polycyclic monoid
We study algebraic structure of the -polycyclic monoid
and its topologizations. We show that the -polycyclic monoid for an
infinite cardinal has similar algebraic properties so has
the polycyclic monoid with finitely many generators. In
particular we prove that for every infinite cardinal the polycyclic
monoid is a congruence-free combinatorial -bisimple
--unitary inverse semigroup. Also we show that every non-zero element
is an isolated point in for every Hausdorff topology
on , such that is a semitopological
semigroup, and every locally compact Hausdorff semigroup topology on
is discrete. The last statement extends results of the paper [33]
obtaining for topological inverse graph semigroups. We describe all feebly
compact topologies on such that
is a semitopological semigroup and its Bohr
compactification as a topological semigroup. We prove that for every cardinal
any continuous homomorphism from a topological semigroup
into an arbitrary countably compact topological semigroup is
annihilating and there exists no a Hausdorff feebly compact topological
semigroup which contains as a dense subsemigroup
On monoids of monotone injective partial self-maps of integers with cofinite domains and images
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 .Comment: arXiv admin note: text overlap with arXiv:1006.487
Sandwich semigroups in locally small categories I: Foundations
Fix (not necessarily distinct) objects and of a locally small
category , and write for the set of all morphisms . Fix a
morphism , and define an operation on by
for all . Then is a
semigroup, known as a sandwich semigroup, and denoted by . This
article develops a general theory of sandwich semigroups in locally small
categories. We begin with structural issues such as regularity, Green's
relations and stability, focusing on the relationships between these properties
on and the whole category . We then identify a natural condition
on , called sandwich regularity, under which the set Reg of all
regular elements of is a subsemigroup of . Under this
condition, we carefully analyse the structure of the semigroup Reg,
relating it via pullback products to certain regular subsemigroups of
and , and to a certain regular sandwich monoid defined on a subset of
; among other things, this allows us to also describe the
idempotent-generated subsemigroup of . We also
study combinatorial invariants such as the rank (minimal size of a generating
set) of the semigroups , Reg and ;
we give lower bounds for these ranks, and in the case of Reg and
show that the bounds are sharp under a certain condition
we call MI-domination. Applications to concrete categories of transformations
and partial transformations are given in Part II.Comment: 23 pages, 1 figure. V2: updated according to referee report, expanded
abstract, to appear in Algebra Universali
Conjugacy in inverse semigroups
The first and second authors were partially supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2019 (Centro de Matematica e Aplicacoes), the project PTDC/MHC-FIL/2583/2014, the FCT project PTDC/MAT-PUR/31174/2017.
The second author was also partially supported by a Simons Foundation Collaboration Grant 359872.In a group G, elements a and b are conjugate if there exists g∈G such that g−1ag=b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b in an inverse semigroup S, a is conjugate to b, which we will write as a∼ib, if there exists g∈S1 such that g−1ag=b and gbg−1=a. The purpose of this paper is to study the conjugacy ∼i in several classes of inverse semigroups: symmetric inverse semigroups, McAllister P-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid, stable inverse semigroups, and free inverse semigroups.publishersversionpublishe
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