Semigroups Of Type Research Articles

On irreducibility and positivity improvement of norm continuous quantum Markov semigroups

We show that for a norm continuous quantum Markov semigroup irreducibility and positivity improvement for any time [Formula: see text] are equivalent under some general conditions that hold, in particular, in the finite-dimensional case. This result is applied to quantum Markov semigroups of weak coupling limit type. We also discuss the discrete-time case.

Numerical semigroups of small and large type

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number [Formula: see text], genus [Formula: see text] and type [Formula: see text]. It is known that for any numerical semigroup [Formula: see text]. Numerical semigroups with [Formula: see text] are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with [Formula: see text]. We show that for a fixed [Formula: see text] the number of numerical semigroups with Frobenius number [Formula: see text] and type [Formula: see text] is eventually constant for large [Formula: see text]. The number of numerical semigroups with genus [Formula: see text] and type [Formula: see text] is also eventually constant for large [Formula: see text].

Homogeneous numerical semigroups

We introduce the concept of homogeneous numerical semigroups and show that all homogeneous numerical semigroups with Cohen-Macaulay tangent cones are of homogeneous type. In embedding dimension three, we classify all numerical semigroups of homogeneous type in to numerical semigroups with complete intersection tangent cones and the homogeneous ones with Cohen-Macaulay tangent cones. We also study the behavior of the homogeneous property by gluing and shiftings to construct a large family of homogeneous numerical semigroups with Cohen-Macaulay tangent cones. In particular we show that these properties fulfill assymptotically in the shifting classes. Several explicit examples are provided along the paper to illustrate the property.

Left and right negatively orderable semigroups and a one-sided version of Simon’s theorem

We study the classes \(\mathrm {LNO}\) and \(\mathrm {RNO}\) of left and right negatively orderable semigroups, arising as natural one-sided generalizations of negatively orderable semigroups (\(\mathrm {NO}\)). Negatively ordered monoids are well-known, for instance, from an equivalent formulation by Straubing and Therien, of a celebrated theorem by I. Simon on piecewise testable languages. The main aim of this paper is to prove a one-sided version of Simon’s theorem for \(\mathrm {LNO}\) (\(\mathrm {RNO}\)). Analogues of some well-known results about negatively orderable semigroups are established in these cases. A characterisation of right negatively orderable semigroups as semigroups of certain type of decreasing mappings on partially ordered sets is obtained. We present sets of quasi-identities defining the quasivarieties \(\mathrm {LNO}\) and \(\mathrm {RNO}\) and show that these classes cannot be defined by finite sets of quasi-identities. We prove \(\mathrm {NO}=\mathrm {LNO}\cap \mathrm {RNO}\) and describe \(\mathrm {LNO}\vee \mathrm {RNO}\). As a one-sided version of Simon’s theorem, the pseudovariety generated by all finite semigroups in \(\mathrm {LNO}\) (\(\mathrm {RNO}\)) is determined and an Eilenberg-type correspondence between this pseudovariety and a variety of languages is established.

A structure theorem for left restriction semigroups of type F

A structure theorem for right adequate semigroups that are F-right abundant otherwise known as right adequate semigroups of type F was given by Cui and Guo. Here we obtain an analogous structure theorem for that of left restriction semigroups: a semigroup is a left restriction semigroup of type F if and only if it is isomorphic to some ℱ(𝑈,𝑌), where (𝑈,𝑌) is an ℱ-pair.

On the structure of quantum Markov semigroups of weak coupling limit type

We discuss some recent results on the structure of quantum Markov semigroups of weak coupling limit type and their stationary states. In particular, we identify the minimal central projection in the fixed point algebra where they act in a trivial way and show, when they admit a single Bohr frequency, that all invariant states are convex combinations of equilibrium states and arbitrary states supported in the above minimal central projection.

A Structure of Some Abundant Semigroups

Semigroup theory is widely applied in many disciplines such as computer science, operation theory and combinatoric mathematics etc.In this paper, we introduce the concept of perfect ♯-abundant semigroups of type C and we overcome the difficulty about results on ♯-abundant which is a little than abundant semigroups,A construction theorem for such semigroups will be given.

The genus, Frobenius number and pseudo-Frobenius numbers of numerical semigroups of type 2

We study some questions on numerical semigroups of type 2. On the one hand, we investigate the relation between the genus and the Frobenius number. On the other hand, for two fixed positive integers g1, g2, we give necessary and sufficient conditions in order to have a numerical semigroup S such that {g1, g2} is the set of its pseudo-Frobenius numbers and, moreover, we explicitly build families of such numerical semigroups.

Topological aperiodicity for product systems over semigroups of Ore type

We prove a version of uniqueness theorem for Cuntz-Pimsner algebras of discrete product systems over semigroups of Ore type. To this end, we introduce Doplicher-Roberts picture of Cuntz-Pimsner algebras, and the semigroup dual to a product system of 'regular' C*-correspondences. Under a certain aperiodicity condition on the latter, we obtain the uniqueness theorem and a simplicity criterion for the algebras in question. These results generalize the corresponding ones for crossed products by discrete groups, due to Archbold and Spielberg, and for Exel's crossed products, due to Exel and Vershik. They also give interesting conditions for topological higher rank graphs and $P$-graphs, and apply to the new Cuntz C*-algebra $\mathcal{Q}_\mathbb{N}$ arising from the "$ax+b$"-semigroup over $\mathbb{N}$.

Parabolic type semigroups: asymptotics and order of contact

We study the asymptotic behavior of parabolic type semigroups acting on the unit disk as well as those acting on the right half-plane. We use the asymptotic behavior to investigate the local geometry of the semigroup trajectories near the boundary Denjoy–Wolff point. The geometric content includes, in particular, the asymptotes to trajectories, the so-called limit curvature, the order of contact, and so on. We then establish asymptotic rigidity properties for a broad class of semigroups of parabolic type.

GROUPS OF FIBONACCI TYPE REVISITED

This article concerns a class of groups of Fibonacci type introduced by Johnson and Mawdesley that includes Conway's Fibonacci groups, the Sieradski groups, and the Gilbert–Howie groups. This class of groups provides an interesting focus for developing the theory of cyclically presented groups and, following questions by Bardakov and Vesnin and by Cavicchioli, Hegenbarth, and Repovš, they have enjoyed renewed interest in recent years. We survey results concerning their algebraic properties, such as isomorphisms within the class, the classification of the finite groups, small cancellation properties, abelianizations, asphericity, connections with Labeled Oriented Graph groups, and the semigroups of Fibonacci type. Further, we present a new method of proving the classification of the finite groups that deals with all but three groups.

Boundary behavior and rigidity of semigroups of holomorphic mappings

In this paper we give some quantitative characteristics of boundary asymptotic behavior of semigroups of holomorphic self-mappings of the unit disk including the limit curvature of their trajectories at the boundary Denjoy–Wolff point. This enable us to establish an asymptotic rigidity property for semigroups of parabolic type.

On Incidence Algebras of Combinatorial Inverse Monoids

We study the incidence algebra of the reduced standard division category of a combinatorial bisimple inverse monoid [with (E(S), ≤) locally finite], and we describe semigroups of poset type (i.e., a combinatorial inverse semigroup for which the corresponding Möbius category is a poset) as being combinatorial strict inverse semigroups. Up to isomorphism, the only Möbius-division categories are the reduced standard division categories of combinatorial inverse monoids.

Local unitary cocycles of [formula omitted]-semigroups

This paper concerns the structure of the group of local unitary cocycles, also called the gauge group, of an E 0 -semigroup. The gauge group of a spatial E 0 -semigroup has a natural action on the set of units by operator multiplication. Arveson has characterized completely the gauge group of E 0 -semigroups of type I, and as a consequence it is known that in this case the gauge group action is transitive. In fact, if the semigroup has index k, then the gauge group action is transitive on the set of ( k + 1 ) -tuples of appropriately normalized independent units. An action of the gauge group having this property is called ( k + 1 ) -fold transitive. We construct examples of E 0 -semigroups of type II and index 1 which are not 2-fold transitive. These new examples also illustrate that an E 0 -semigroup of type II k need not be a tensor product of an E 0 -semigroup of type II 0 and another of type I k .

A perturbation problem for the shift semigroup

Motivated by B. Tsirelson's construction of E 0 -semigroups of type III, we investigate a C 0 -semigroup acting on the space of square integrable functions of the half line whose difference from the shift semigroup is a Hilbert–Schmidt operator. We give a description of such semigroups in terms of analytic functions on the right-half plane and construct several examples.

Left semi-perfect abundant semi-groups of type W

We call a quasi-adequate semi-group whose set of idempotents forms a left [right] quasi-normal band a left [right] semi-perfect abundant semi-group. After obtaining some characterization theorems of such quasi-adequate semi-groups, we establish a structure for left [right] semi-perfect abundant semi-groups of type W. Our results generalize and strengthen the results of El-Qallali and Fountain on quasi-adequate semi-groups.

Left semi-perfect abundant semi-groups of type W

We call a quasi-adequate semi-group whose set of idempotents forms a left [right] quasi-normal band a left [right] semi-perfect abundant semi-group. After obtaining some characterization theorems of such quasi-adequate semi-groups, we establish a structure for left [right] semi-perfect abundant semi-groups of type W. Our results generalize and strengthen the results of El-Qallali and Fountain on quasi-adequate semi-groups.

THE IMMEDIATE BASIN OF ATTRACTION OF INFINITY FOR POLYNOMIAL SEMIGROUPS OF FINITE TYPE

As a generalization of the dynamics of a single polynomial, Hinkkanen and Martin studied the dynamics of polynomial semigroups of finite type. Among the issues addressed were possible generalizations of the filled-in Julia set of a single polynomial and the basin of attraction for infinity. In the paper, these concepts are further refined, and the connections between the dynamics of a single function and the dynamics of semigroups are strengthened.

An Index Theory For Quantum Dynamical Semigroups

W. Arveson showed a way of associating continuous tensor product systems of Hilbert spaces with endomorphism semigroups of type I factors. We do the same for general quantum dynamical semigroups through a dilation procedure. The product system so obtained is the index and its dimension is a numerical invariant for the original semigroup.

Weakly uniform completely O-simple semigroups of matrix type

Weakly uniform completely O-simple semigroups of matrix type