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.     

Restriction semigroups and $$\lambda $$-Zappa-Szép products

The aim of this paper is to study \(\lambda \)-semidirect and \(\lambda \)-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product \(S\bowtie T\) where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-Schein-Nambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are \(\lambda \)-semidirect products and \(\lambda \)-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of \(\lambda \)-semidirect products.     

Algebraic properties of Zappa–Szép products of semigroups and monoids

Direct, semidirect and Zappa–Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa–Szép products of monoids and semigroups and investigate generalised Greens relations \({\mathcal R}^{*},\, {\mathcal L}^{*},\, \widetilde{\mathcal {R}}_E\) and \(\widetilde{\mathcal {L}}_E\) for these Zappa–Szép products. We consider a left restriction semigroup S with semilattice of projections E and define left and right actions of S on E and E on S, respectively, to form the Zappa–Szép product \(E \bowtie S\). We further investigate properties of \(E \bowtie S\) and show that S is a retract of \(E\bowtie S\). We also find a subset T of \(E \bowtie S\) which is left restriction.     

Weyl–Pedersen calculus for some semidirect products of nilpotent Lie groups

For certain nilpotent real Lie groups constructed as semidirect products, algebras of invariant differential operators on some coadjoint orbits are used in the study of boundedness properties of the Weyl–Pedersen calculus of their corresponding unitary irreducible representations. Our main result is applicable to all unitary irreducible representations of arbitrary 3-step nilpotent Lie groups.     

Stability for Lie–Trotter products for some operator matrix semigroups

Coupled systems of linear differential equations in Banach spaces can be often handled by the theory of C₀-semigroups of operator matrices. We study the stability of Lie–Trotter products of such matrix semigroups, and present three classes of examples (abstract delay equations, abstract inhomogeneous equations, abstract dynamic boundary value problems) and some open problems. This survey is based on the papers [1], [2] and [5], to which we refer the interested reader for more details and extensive bibliographical information. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bruck–Reilly extensions of direct products of monoids and completely (0-)simple semigroups

We say that a class of monoids satisfies the property ℘ if every monoid in that class that admits a finitely presented Bruck–Reilly extension is finitely generated. We show that completely (0-)simple semigroups satisfy ℘, and that the direct product of two monoids in a class that satisfy ℘ also satisfies ℘ subject to a certain condition on the endomorphisms of the direct product. As a consequence of this result we obtain a new class of bands and a new class of completely regular semigroups that satisfy property ℘.     

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.     

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.     

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.     

Comparison semigroups and algebras of transformations

We characterize algebras of transformations on a set under the operations of composition and the pointwise switching function defined as follows: (f,g)[h,k](x)=h(x) if f(x)=g(x), and k(x) otherwise. The resulting algebras are both semigroups and comparison algebras in the sense of Kennison. The same characterization holds for partial transformations under composition and a suitable generalisation of the quaternary operation in which agreement of f,g includes cases where neither is defined. When a zero element is added (modelling the empty function), the resulting signature is rich enough to encompass many operations on semigroups of partial transformations previously considered, including set difference and intersection, restrictive product, and a functional analog of union. When an identity element is also added (modelling the identity function), further domain-related operations can be captured.     

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.     

Hyperbolic operator semigroups and Lyapunov’s equation

We obtain necessary and sufficient conditions for the hyperbolicity of a semigroup of operators. In so doing, we use Lyapunov’s equation in operator form constructed from its generator.     

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.     

Joint pricing and procurement of substitutable products with random demands – A technical note

Single-period joint pricing and procurement of substitutable products entails one time procurement and pricing decisions for substitutable products that face price dependent stochastic demands. Recently, Karakul and Chan [Karakul, M., Chan, L., 2008. Analytical and managerial implications of integrating product substitutability in the joint pricing and procurement problem. European Journal of Operational Research 190, 179–204] considered this problem for two one-way substitutable products. Authors model the demands for each product in the well known additive form, where the mean demands are linear functions of the price of the high grade new product plus an additive stochastic noise term. By assuming that the noise term for the low grade existing product follows a general discrete distribution and the noise term for the high grade product follows a general continuous distribution, authors are able to show the unimodality of the expected profit function with respect to the procurement quantities and the price of the new product. In this paper, we extend this result to the case where the noise term in the demand of the low grade product follows a general continuous distribution as well.     

η-Simple semigroups without zero and η ∗-simple semigroups with a least non-zero idempotent

A semigroup S is called η-simple if S has no semilattice congruences except S×S. Tamura in (Semigroup Forum 24:77–82, 1982) studied η-simple semigroups with a unique idempotent. In the present paper we consider a more general situation, that is, we investigate η-simple semigroups (without zero) with a least idempotent. Moreover, we study η ^?-simple semigroups with zero which contain a least non-zero idempotent.     

Semidirect products and wreath products of strongly π-inverse monoids

In this paper we determine the necessary and sufficient conditions for the semidirect products and the wreath products of two monoids to be strongly -inverse. Furthermore, we determine the least group congruence on a strongly -inverse monoid, and we give some important isomorphism theorems.