Morita invariants for semigroups with local units

In this paper we study Morita invariants for strongly Morita equivalent semigroups with local units of various kinds. Among others we prove that, under a certain condition of this kind, congruence lattices are preserved by strong Morita equivalence.     

Morita equivalence of semigroups with local units

We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister’s theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locally inverse semigroup.     

Strong Morita equivalence for ordered semigroups with local units

We prove that partially ordered semigroups with local units are strongly Morita equivalent if and only if they have a joint enlargement, which in turn happens if and only if the Cauchy completions of the semigroups are equivalent.     

Strong Morita equivalence of semigroups with local units

In this paper we study Morita contexts for semigroups. We prove a Rees matrix cover connection between strongly Morita equivalent semigroups and investigate how the existence of a unitary Morita semigroup over a given semigroup is related to the existence of a ‘good’ Rees matrix cover of this semigroup.     

Automorphism invariants for semigroups

Invariant subgroups are associated with each element of a semigroup. These invariants are used to show certain semigroups of continuous functions have only inner automorphisms. In special cases bijections preserving these invariants are necessarily automorphisms and outer automorphisms can be constructed.     

正则系对有局部单位半群的刻画

借助酉系范畴中的正则对象研究了有局部单位半群的同调分类问题.通过研究正则对象中元素的性质,建立了正则性质与余直积以及正合序列之间的联系;通过探索正则对象与主弱平坦对象以及投射对象的关系,给出了有局部单位半群同调分类的刻画.     

Monoidal Morita invariants for finite group algebras

Two Hopf algebras are called monoidally Morita equivalent if module categories over them are equivalent as linear monoidal categories. We introduce monoidal Morita invariants for finite-dimensional Hopf algebras based on certain braid group representations arising from the Drinfeld double construction. As an application, we show, for any integer n, the number of elements of order n is a monoidal Morita invariant for finite group algebras. We also describe relations between our construction and invariants of closed 3-manifolds due to Reshetikhin and Turaev.     

Morita equivalence of finite semigroups

Semigroup Forum - Two semigroups are called Morita equivalent if the categories of firm right acts over them are equivalent. We prove that every semigroup is Morita equivalent to its subsemigroup...     

Morita equivalence of inverse semigroups

We describe how to construct all inverse semigroups Morita equivalent to a given inverse semigroup S. This is done by taking the maximum inverse images of the regular Rees matrix semigroups over S where the sandwich matrix satisfies what we call the McAlister conditions.     

Characterizations of Morita equivalent inverse semigroups

We prove that four different notions of Morita equivalence for inverse semigroups motivated by C^∗-algebra theory, topos theory, semigroup theory and the theory of ordered groupoids are equivalent. We also show that the category of unitary actions of an inverse semigroup is monadic over the category of étale actions. Consequently, the category of unitary actions of an inverse semigroup is equivalent to the category of presheaves on its Cauchy completion. More generally, we prove that the same is true for the category of closed actions, which is used to define the Morita theory in semigroup theory, of any semigroup with right local units.     

Conditions for strong Morita equivalence of partially ordered semigroups

We investigate when a partially ordered semigroup (with various types of local units) is strongly Morita equivalent to a posemigroup from a given class of partially ordered semigroups. Necessary and sufficient conditions for such equivalence are obtained for a series of well-known classes of posemigroups. A number of sufficient conditions for several classes of naturally ordered posemigroups are also provided.     

Generalized heaps,inverse semigroups and Morita equivalence

Inverse semigroups are the algebraic counterparts of pseudogroups of transformations. The algebraic counterparts of atlases in differential geometry are what Wagner termed ‘generalized heaps’. These are sets equipped with a ternary operation satisfying certain axioms. We prove that there is a bijective correspondence between generalized heaps and the equivalence bimodules, defined by Steinberg. Such equivalence bimodules are used to define the Morita equivalence of inverse semigroups. This paper therefore shows that the Morita equivalence of inverse semigroups is determined by Wagner’s generalized heaps.     

Topological invariants for semigroups of holomorphic self-maps of the unit disk

Let $({\phi }_t)$, $({?}_t)$ be two one-parameter semigroups of holomorphic self-maps of the unit disk $\mathbb{D}?\mathbb{C}$. Let $f:\mathbb{D}\to \mathbb{D}$ be a homeomorphism. We prove that, if $f°{?}_t={\phi }_t°f$ for all $t\ge 0$, then f extends to a homeomorphism of $\stackrel{￣}{\mathbb{D}}$ outside exceptional maximal contact arcs (in particular, for elliptic semigroups, f extends to a homeomorphism of $\stackrel{￣}{\mathbb{D}}$). Using this result, we study topological invariants for one-parameter semigroups of holomorphic self-maps of the unit disk.     

Separating invariants and local cohomology

The study of separating invariants is a recent trend in invariant theory. For a finite group acting linearly on a vector space, a separating set is a set of invariants whose elements separate the orbits of G. In some ways, separating sets often exhibit better behavior than generating sets for the ring of invariants. We investigate the least possible cardinality of a separating set for a given G-action. Our main result is a lower bound that generalizes the classical result of Serre that if the ring of invariants is polynomial then the group action must be generated by pseudoreflections. We find these bounds to be sharp in a wide range of examples.