Matrix semigroups—generators and relations

Some presentations for the semigroups of all 2×2 matrices and all 2×2 matrices of determinant 0 or 1 over the field GF(p) (p prime) are given. In particular, if <a, b, c‖ R> is any (semigroup) presentation for the general linear group in terms of generators $$A = \left( {\begin{array}{*{20}c} 1 & 0 \\ 1 & 1 \\ \end{array} } \right),B = \left( {\begin{array}{*{20}c} 1 & 1 \\ 0 & 1 \\ \end{array} } \right),C = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & \xi \\ \end{array} } \right),$$ where ζ is a primitive root of 1 modulop, then the presentation $$\langle a,b,c,t|R,t^2 = ct = tc = t,tba^{p - 1} t = 0,b^{\xi - 1} atb = a^{\xi - 1} tb^\xi a^{1 - \xi - 1} \rangle $$ defines the semigroup of all 2×2 matrices over GF (2,p) in terms of generatorsA, B, C and $$T = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & 0 \\ \end{array} } \right).$$ Generating sets and ranks of various matrix semigroups are also found.     

Möbius and triangle semigroups

The purpose of this paper is to describe the structures of the Möbius semigroup induced by the Möbius transformation group (?, SL(2,?)). In particular, we study stabilizer subsemigoups of Möbius semigroup via the triangle semigroup. In this work, we obtained a geometric interpretation of the least contraction coefficient function of the Möbius semigroup via the triangle semigroup and investigated an extension of stabilizer subsemigoups of the Möbius semigroup. Finally, we obtained a factorization of our stabilizer subsemigoups of the Möbius semigroup.     

Dunkl-Schrödinger semigroups and applications

We obtain dispersive estimates for the linear Dunkl–Schrödinger equations with and without quadratic potential. As a consequence, we prove the local well-posedness for semilinear Dunkl–Schrödinger equations with polynomial nonlinearity in certain magnetic field. Furthermore, we study many applications: as the uncertainty principles for the Dunkl transform via the Dunkl–Schrödinger semigroups, the embedding theorems for the Sobolev spaces associated with the generalized Hermite semigroup. Finally, almost every where convergence of the solutions of the Dunkl–Schrödinger equation is also considered.     

The structure of L*-inverse semigroups

The concepts of L*-inverse semigroups and left wreath products of semigroups are introduced. It is shown that the L*-inverse semigroup can be described as the left wreath product of a type A semigroupΓand a left regular band B together with a mapping which maps the semigroupΓinto the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups. We shall also provide a constructed example for the L*-inverse semigroups by using the left wreath products.     

Perturbation theorems for α-times integrated semigroups

We prove perturbation results for -times integrated semigroups assuming relative smallness conditions for the perturbation B on a halfplane. If A is a semigroup generator on a uniformly convex Banach space, then these conditions on B already imply that A + B generates a once integrated semigroup. As an illustration we consider Schrödinger operators and higher order differential operators.Received: 30 April 2002     

Generalized inverses modulo ℋ in semigroups and rings

The definition of the inverse along an element was very recently introduced, and it contains known generalized inverses such as the group, Drazin and Moore–Penrose inverses. In this article, we first prove a simple existence criterion for this inverse in a semigroup, and then relate the existence of such an inverse in a ring to the ring units.     

Erdős-Ginzburg-Ziv theorem for finite commutative semigroups

Let \(\mathcal{S}\) be a finite additively written commutative semigroup, and let \(\exp(\mathcal{S})\) be its exponent which is defined as the least common multiple of all periods of the elements in \(\mathcal{S}\) . For every sequence T of elements in \(\mathcal{S}\) (repetition allowed), let \(\sigma(T) \in\mathcal{S}\) denote the sum of all terms of T. Define the Davenport constant \(\mathsf{D}(\mathcal{S})\) of \(\mathcal{S}\) to be the least positive integer d such that every sequence T over \(\mathcal{S}\) of length at least d contains a proper subsequence T′ with σ(T′)=σ(T), and define \(\mathsf{E}(\mathcal{S})\) to be the least positive integer ? such that every sequence T over \(\mathcal{S}\) of length at least ? contains a subsequence T′ with \(|T|-|T'|= \lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})} \rceil \exp(\mathcal{S})\) and σ(T′)=σ(T). When \(\mathcal{S}\) is a finite abelian group, it is well known that \(\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})} \rceil\exp (\mathcal{S})=|\mathcal{S}|\) and \(\mathsf{E}(\mathcal{S})=\mathsf{D}(\mathcal{S})+|\mathcal{S}|-1\) . In this paper we investigate whether \(\mathsf{E}(\mathcal{S})\leq \mathsf{D}(\mathcal{S})+ \lceil\frac{|\mathcal{S}|}{\exp(\mathcal {S})} \rceil \exp(\mathcal{S})-1\) holds true for all finite commutative semigroups \(\mathcal{S}\) . We provide a positive answer to the question above for some classes of finite commutative semigroups, including group-free semigroups, elementary semigroups, and archimedean semigroups with certain constraints.     

Perturbations and projections of Kalman–Bucy semigroups

We analyse various perturbations and projections of Kalman–Bucy semigroups and Riccati equations. For example, covariance inflation-type perturbations and localisation methods (projections) are common in the ensemble Kalman filtering literature. In the limit of these ensemble methods, the regularised sample covariance tends toward a solution of a perturbed/projected Riccati equation. With this motivation, results are given characterising the error between the nominal and regularised Riccati flows and Kalman–Bucy filtering distributions. New projection-type models are also discussed; e.g. Bose–Mesner projections. These regularisation models are also of interest on their own, and in, e.g., differential games, control of stochastic/jump processes, and robust control.     

Orthogonal sums of 0-σ-simple semigroups

Supported by Grant 0401 A of RFNS through Math. Inst. SANU.     

Proper restriction semigroups – semidirect products and W-products

Fountain and Gomes [4] have shown that any proper left ample semigroup embeds into a so-called W-product, which is a subsemigroup of a reverse semidirect product ${T\ltimes {\mathcal {Y}}}$ of a semilattice ${\mathcal {Y}}$ by a monoid T, where the action of T on  ${\mathcal {Y}}$ is injective with images of the action being order ideals of  ${\mathcal {Y}}$ . Proper left ample semigroups are proper left restriction, the latter forming a much wider class. The aim of this paper is to give necessary and sufficient conditions on a proper left restriction semigroup such that it embeds into a W-product. We also examine the complex relationship between W-products and semidirect products of the form ${{\mathcal {Y}}\rtimes T}$ .     

On Kovács–Newman ordered semigroups

Semigroup Forum - The irreducibility of pseudovarieties of semigroups is studied using many different approaches including a notion of Kovács–Newman semigroups. In this note, we answer a...     

The Möbius function on Abelian semigroups

Let X be an Abelian semigroup such that the following conditions hold: (i) if x × y = II (II is the identity element), then x = y = II; (ii) the set {{x, y}: x × y = a} is finite for any a ∈ X. Let Λ be any field, and let ℰ be the algebra of all Λ-valued functions on X. The convolution of u, υ ∈ ℰ is defined by

Automatic structures for subsemigroups of Baumslag–Solitar semigroups

This paper studies automatic structures for subsemigroups of Baumslag–Solitar semigroups (that is, semigroups presented by 〈x,y∣(yx ^m ,x ⁿ y)〉 where $m,n \in\mathbb {N}$ ). A geometric argument (a rarity in the field of automatic semigroups) is used to show that if m>n, all of the finitely generated subsemigroups of this semigroup are (right-) automatic. If m<n, all of its finitely generated subsemigroups are left-automatic. If m=n, there exist finitely generated subsemigroups that are not automatic. An appendix discusses the implications of these results for the theory of Malcev presentations. (A Malcev presentation is a special type of presentation for semigroups embeddable into groups.)     

∗-congruences on abundant semigroups with SQ-adequate transversals

We give ?-congruences on an abundant semigroup with an SQ-adequate transversal S ^° by the ?-congruence triple abstractly which consists of congruences on the structure component parts L, T and R. We prove that the set of all ?-congruences on this kind of semigroups is a complete lattice.     

D-saturated property of the Cayley graphs of semigroups

Let D be a finite graph. A semigroup S is said to be Cayley D-saturated with respect to a subset T of S if, for all infinite subsets V of S, there exists a subgraph of Cay(S,T) isomorphic to D with all vertices in V. The purpose of this paper is to characterize the Cayley D-saturated property of a semigroup S with respect to any subset T⊆S. In particular, the Cayley D-saturated property of a semigroup S with respect to any subsemigroup T is characterized.     

Quasi-ideal transversals of abundant semigroups and spined products

We provide a new and much simpler structure for quasi-ideal adequate transversals of abundant semigroups in terms of spined products, which is similar in nature to that given by Saito in Proc. 8th Symposium on Semigroups, pp. 22–25 (1985) for weakly multiplicative inverse transversals of regular semigroups. As a consequence we deduce a similar result for multiplicative transversals of abundant semigroups and also consider the case when the semigroups are in fact regular and provide some new structure theorems for inverse transversals.