EMBEDDING THEOREMS FOR GROUPS PRESENTED VIA PARTIAL AUTOMORPHISMS: A GENERALIZATION OF SEMIDIRECT PRODUCTS AND HNN EXTENSIONS

A partial automorphism of a semigroup $S$ is any isomorphism between its subsemigroups, and the set all partial automorphisms of $S$ with respect to composition is the inverse monoid called the partial automorphism monoid of $S$. Two semigroups are said to be $\cPA$-isomorphic if their partial automorphism monoids are isomorphic. A class $\K$ of semigroups is called $\cPA$-closed if it contains every semigroup $\cPA$-isomorphic to some semigroup from $\K$. Although the class of all inverse semigroups is not $\cPA$-closed, we prove that the class of inverse semigroups, in which no maximal isolated subgroup is a direct product of an involution-free periodic group and the two-element cyclic group, is $\cPA$-closed. It follows that the class of all combinatorial inverse semigroups (those with no nontrivial subgroups) is $\cPA$-closed. A semigroup is called $\cPA$-determined if it is isomorphic or anti-isomorphic to any semigroup that is $\cPA$-isomorphic to it. We show that combinatorial inverse semigroups which are either shortly connected [5] or quasi-archimedean [10] are $\cPA$-determined.

A discrete group which admits a faithful, finite dimensional, linear representation over a field F of characteristic zero is called linear.This note combines the natural structure of semi-direct products with work of A. Lubotzky [13] on the existence of linear representations to develop a technique to give sufficient conditions to show that a semi-direct product is linear.Let G denote a discrete group which is a semi-direct product given by a split extensionThis note defines an additional type of structure for this semi-direct product called a stable extension below.The main results are as follows:(1) If π and Γ are linear, and the extension is stable, then G is also linear.Restrictions concerning this extension are necessary to guarantee that G is linear as seen from properties of the Formanek-Procesi "poison group" [7].(2) If the action of Γ on π has a "Galois-like" property that it factors through the automorphisms of certain natural "towers of groups over π" ( to be defined below ), then the associated extension is stable and thus G is linear.(3) The condition of a stable extension also implies that G admits filtration quotients which themselves give a natural structure of Lie algebra and

Embedding theorems for HNN extensions of inverse semigroups

Sur des représentations indécomposables de dimension finie de SL(2).R 2

The aim of this paper is to study \(\lambda \)-semidirect and \(\lambda \)-Zappa-Szep 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-Szep 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-Szep products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of \(\lambda \)-semidirect products.

Rees Matrix Covers and Semidirect Products of Regular Semigroups

We consider semigroups such that the universal left congruence omega ^{ell } is finitely generated. Certainly a left noetherian semigroup, that is, one in which all left congruences are finitely generated, satisfies our condition. In the case of a monoid the condition that omega ^{ell } is finitely generated is equivalent to a number of pre-existing notions. In particular, a monoid satisfies the homological finiteness condition of being of type left-{text {FP}}_1 exactly when omega ^{ell } is finitely generated.Our investigations enable us to classify those semigroups such that omega ^{ell } is finitely generated that lie in certain important classes, such as strong semilattices of semigroups, inverse semigroups, Rees matrix semigroups (over semigroups) and completely regular semigroups. We consider closure properties for the class of semigroups such that omega ^{ell } is finitely generated, including under morphic image, direct product, semi-direct product, free product and 0-direct union. Our work was inspired by the stronger condition, stated for monoids in the work of White, of being pseudo-finite. Where appropriate, we specialise our investigations to pseudo-finite semigroups and monoids. In particular, we answer a question of Dales and White concerning the nature of pseudo-finite monoids.

E-unitary inverse monoids and the Cayley graph of a group presentation

We compare the two recently introduced semidirect product operations * r and * rr within the lattice of e-varieties of locally inverse semigroups. For each e-variety $$ \mathcal{V}$$ which contains all rectangular bands and is properly contained in the e-variety of all completely simple semigroups, the inclusions $$ \mathcal{S} *_{rr} \mathcal{V} \subsetneq \mathcal {S} *_r \mathcal{V} \subsetneq \mathcal{S} *_{rr} (\mathcal{A}_q \circ \mathcal{V}) $$ are proved where $$ \mathcal{S} $$ is the e-variety of all semilattices and $$ \mathcal{A}_q $$ the variety of all abelian groups of exponent dividing q where q is any integer greater than one. Some consequences for the class of finite locally inverse semigroups are also obtained.

We study automaton structures, i.e., groups, monoids and semigroups generated by an automaton, which, in this context, means a deterministic finite-state letter-to-letter transducer. Instead of considering only complete automata, we specifically investigate semigroups generated by partial automata. First, we show that the class of semigroups generated by partial automata coincides with the class of semigroups generated by complete automata if and only if the latter class is closed under removing a previously adjoined zero, which is an open problem in (complete) automaton semigroup theory stated by Cain. Then, we show that no semidirect product (and, thus, also no direct product) of an arbitrary semigroup with a (non-trivial) subsemigroup of the free monogenic semigroup is an automaton semigroup. Finally, we concentrate on inverse semigroups generated by invertible but partial automata, which we call automaton-inverse semigroups, and show that any inverse automaton semigroup can be generated by such an automaton (showing that automaton-inverse semigroups and inverse automaton semigroups coincide).

A normal Hall subgroup $N$ of a group $G$ is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall [STACS 2009], we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai. Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations rho and tau of a group $H$ over Z_p^d , p a prime, determine if there exists an automorphism phi:H -> H, such that the induced representation rho_phi=rho o phi and tau are equivalent, in time poly(|H|,p^d).

Let G be a connected Lie group which is a semidirect product of a compact subgroup K and a normal solvable subgroup S. Let Λ \Lambda be a character of S, and let M Λ {M_\Lambda } be the stabilizer of Λ \Lambda in K. Let [ H , Λ μ ] [H,{\Lambda _\mu }] be a finite-dimensional irreducible representation of the subgroup S M Λ S{M_\Lambda } on the complex vector space H. In this paper we consider the induced representations of G on various Banach spaces, and study their topological irreducibility. The basic method used consists in studying the irreducibility of the Lie algebra representations which arise on the linear subspaces of K-finite vectors. The latter question then can be reduced to the problem of determining when certain modules over certain commutative algebras are irreducible. The method discussed in this paper leads to two theorems giving sufficient conditions on the character Λ \Lambda that the induced representations be topologically irreducible. The question of infinitesimal equivalence of various induced representations is also discussed.

In present work, in order to avoid the spread of disease, the impulse control strategy is implemented to keep the density of infections at a low level. The SIR epidemic model with resource limitation including a nonlinear impulsive function and a state-dependent feedback control scheme is proposed and analyzed. Based on the qualitative properties of the corresponding continuous system, the existence and stability of positive order-k (kinmathbf{Z}^{+}) periodic solution are investigated. By using the Poincaré map and the geometric method, some sufficient conditions for the existence and stability of positive order-1 or order-2 periodic solution are obtained. Moreover, the sufficient conditions which guarantee the nonexistence of order-k (kgeq3) periodic solution are given. Some numerical simulations are carried out to illustrate the feasibility of our main results.

In an unbalanced, fixed-effects, two-way analysis of variance there are three distinct, frequently encountered hypotheses, each of which can be described as a hypothesis of “no row effects.” This article derives necessary and sufficient conditions on the cell sizes for the “rows adjusted for columns” hypothesis to be identical to the “simple row means” hypothesis. The results in this article, together with previous results, complete the characterizations in terms of cell sizes for pairwise equivalence of the three hypotheses. Geometric methods are used in the proofs.

In this paper, we study a new SEIRS epidemic model describing nonlinear incidence with a more general form and the transmission of influenza virus with disease resistance. The basic reproductive number Re_{0} is obtained by using the method of next generating matrix. If Re_{0}<1, the disease-free equilibrium is globally asymptotically stable, and if Re_{0}>1, by using the geometric method, we obtain some sufficient conditions for global stability of the unique endemic equilibrium. Finally, numerical simulations are provided to support our theoretical results.