A history of topological and analytical semigroups: a personal view

A retrospective of the historical development of a topological and analytical theory of semigroups is given from a personal vantage point. It begins with SOPHUS LIE who from about 1880 onward dealt with semigroups by default, having no clear concept of a group at first. The algebraic theory of semigroups emerged in the first half of the 20th century, but its topological counterpart emancipated itself as late as in the second half. I shall comment on the genesis of a theory of compact topological semigroups in the fifties under the influence of A. D. WALLACE. These semigroups came into focus at about the same time E. S. LYAPIN raised the important issue of magnifying elements, thereby discovering the bicyclic semigroup wherever those exist. Compact topological semigroups, however, cannot contain bicyclic semigroups; this has interesting consequences. - Around 1970 D. S. SCOTT discovered what he called continuous lattices and what nowadays, in more general form, is called domains, whileJ. D. LAWSON drew semigroup theoreticians' attention to a very natural class of compact semilattices having enough homomorphisms into the unit interval semilattice. The class of continuous lattices agrees with the class of Lawson semilattices. It generates a network of applications in theoretical computer science under the name "domain theory". - A hundred years after SOPHUS LIE's differentiable groups and semigroups, attention returned back to semigroups and Lie theory. Lie semigroup theory, initiated by E. B. VINBERG, G. I. OLSHANSKY, J. D. LAWSON and the author among others, infused a strong geometric and analytical flavor into topological semigroup theory and generated a new lines of application of semigroup theory such as in geometric control theory, and in the area of unitary representation theory of Lie groups, particulary in the area of holomorphic extensions of unitary representations. A respectable number of mongraphs and collections have been and are being written in this field.     

Minimal Presentations and Efficiency of Semigroups

A finite semigroup S is said to be efficient if it can be defined by a presentation (A | R) with |R| -|A|=rank(H2(S)). In this paper we demonstrate certain infinite classes of both efficient and inefficient semigroups. Thus, finite abelian groups, dihedral groups D2n with n even, and finite rectangular bands are efficient semigroups. By way of contrast we show that finite zero semigroups and free semilattices are never efficient. These results are compared with some well-known results on the efficiency of groups.     

Spectral Inclusions for Semigroups of Closed Operators

We show that several spectral inclusions known for C0-semigroups fail for semigroups of closed operators, even if they can be regularized. We introduce the notion of spectral completeness for the regularizing operator C which implies equality of the spectrum and the C-spectrum of the generator. We prove spectral inclusions under this additional assumption. We give a series of examples in which the regularizing operator is spectrally complete including generators of integrated semigroups, of distribution semigroups, and of some semigroups that are strongly continuous for t > 0.     

Structural results for semigroup subsets defined by factorization properties dependent on \Omega functions

We study combinatorial properties of a class of subsets of semigroups with divisor theory motivated by the study of oscillations of counting functions of sets of algebraic integers with prescribed factorization properties.     

Cleavability in Semigroups

The notion of cleavability (splittability) is observed to apply not only to topological spaces, where it was first developed, but to semigroups also. Several results of the form 'if a semigroup D is cleavable over a class of semigroups each of which has property P, then D also has property P' are derived, and some suggestions for further investigations are put forward.     

The structure of zero-divisor semigroups with graph K n ○K 2

This paper determines all commutative zero divisor semigroups whose zero divisor graph is a complete graph (finite or infinite), or a complete graph (finite or infinite) with one additional end vertex, and gives formulas for the numbers of all such semigroups with n elements. The research of T. Wu is supported by the National Natural Science Foundation of China (Grant No. 10671122) and the Natural Science Foundation of Shanghai (Grant No. 06ZR14049).     

Oids,finite sums and the structure of the Stone-Čech compactification of a discrete semigroup

The equivalence between oids and semigroups which satisfy a condition involving finite sums is established. Some of the already known results on the structure of Stone-Čech compactifications of discrete semigroups are obtained as immediate consequences. It is also shown that most commutative semigroups contain oids so that oid theory has applications to the Stone-Čech compactifications of many semigroups.     

Generalized divisors on Gorenstein schemes

We develop a theory of generalized divisors on a Gorenstein scheme whereby any closed subscheme of pure codimension one without embedded points can be regarded as an effective divisor. Most of the usual theory of linear equivalence, associated sheaf, etc., carries over to this more general setting. The definition uses reflexive sheaves, so we first review the theory of reflexive modules. As an application, we give new definitions of liaison and biliaison for subschemes of ⁿ , which simplify the foundations of the theory of liaison. We also compute explicitly the set of generalized divisor classes on some reducible and singular schemes.     

Feller Semigroups Generated by Degenerate Elliptic Operators

This paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups with sticky boundary condition on various spaces of continuous functions, generalizing the previous work. More precisely we construct Feller semigroups corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the state space, sticking to the boundary.     

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.     

Context equivalence of semigroups

Two semigroups are called strongly Morita equivalent if they are contained in a Morita context with unitary bi-acts and surjective mappings. We consider the notion of context equivalence which is obtained from the notion of strong Morita equivalence by dropping the requirement of unitariness. We show that context equivalence is an equivalence relation on the class of factorisable semigroups and describe factorisable semigroups that are context equivalent to monoids or groups, and semigroups with weak local units that are context equivalent to inverse semigroups, orthodox semigroups or semilattices.     

Inverse semigroups associated with one-dimensional generalized solenoids

In this paper, we study inverse semigroups defined on the Bratteli–Vershik systems and SFT covers of 1-solenoids. We show that groupoids of germs of these inverse semigroups are equivalent to the unstable equivalence groupoids of 1-solenoids. Then we prove that Exel’s tight \(C^*\)-algebras of inverse semigroups are strongly Morita equivalent to the unstable \(C^*\)-algebras of 1-solenoids.     

A note on orthodox semidirect products and wreath products of monoids

In the present note, we show that the restriction which Saito imposed on S in [3] is not necessary: S may be a semigroup without unity. Secondly, we investigate the wreath products of semigroups and obtain our main results, Theroem 2.3 and Corollary 2.4. We find also that groups in [3] may be replaced by right groups in [1]. Undefined terminology and notation may be found in [3] and [1] unlesss otherwise stated.     

Some Remarks on Additive Arithmetical Semigroups

Under weak conditions called axioms and , we prove Chebyshevtype estimates and asymptotic formulas, respectively, for the prime elements in general additive arithmetical semigroups. As applications, we derive asymptotic laws for the mean behavior of prime divisor functions and of distinct degrees of prime factors.     

Some notes on strongly E*-unitary inverse semigroups

In this paper, we give a direct proof that every strongly inverse semigroup can be embedded into a 0-semidirect product of a semilattice with zero by a group. As a corollary, we obtain a new proof of the structure theory of strongly inverse semigroups described in [1]. We also prove that the strongly inverse semigroups are precisely inverse semigroups equipped with a , idempotent pure prehomomorphism to a primitive inverse semigroup.     

Labelling classes by sets

Let Q be an equivalence relation whose equivalence classes, denoted Q[x], may be proper classes. A function L defined on Field(Q) is a labelling for Q if and only if for all x,L(x) is a set andL is a labelling by subsets for Q if and only ifBG denotes Bernays-Gödel class-set theory with neither the axiom of foundation, AF, nor the class axiom of choice, E. The following are relatively consistent with BG. (1) E is true but there is an equivalence relation with no labelling.(2) E is true and every equivalence relation has a labelling, but there is an equivalence relation with no labelling by subsets.This research was partially supported by Fondecyt 1980855 and by Fondecyt 1040846     

Stratified semigroups

Stratified semigroups include free semigroups, finite nilsemigroups, and homogeneous semigroups (=with homogeneous presentations). Basic properties of this class of semigroups are given, including the role that certain combinatorial structures (homogeneous equivalence relations) play in their construction. After this article was typeset the author discovered that stratified semigroups were also considered by Sasaki and Tamura (Proc. Japan. Acad. Ser. A Math. Phys.63 (1987), 315–317), who called themboundly factorizable.     

A class of semigroups regularized in space and time

We consider α-times integrated C-regularized semigroups, which are a hybrid between semigroups regularized in space (C-semigroups) and integrated semigroups regularized in time. We study the basic properties of these objects, also in absence of exponential boundedness. We discuss their generators and establish an equivalence theorem between existence of integrated regularized semigroups and well-posedness of certain Cauchy problems. We investigate the effect of smoothing regularized semigroups by fractional integration.     

保等价部分变换半群的变种半群上的正则元

在现有的保等价部分变换半群的基础上,引入了一个新的运算,得出保等价部分变换半群的变种半群的概念,利用格林关系及幂等元的正则性,讨论了这类半群中元素的正则性,给出了保等价部分变换半群的变种半群中一个元是正则元的充要条件     

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.