Representations Of Semigroups Research Articles

Semigroup theory and analysis of solutions for a higher order non‐Lipschitz problem with advection in ℝN$$ {\mathrm{\mathbb{R}}}^N $$

In studying heterogeneous diffusion phenomena beyond the classical order 2 Laplacian, higher order operators emerge as powerful tools. When such operators are applied, the resultant profiles often showcase an oscillatory nature that may be noticeable near the zero stationary solution. These oscillations can pose challenges, especially when they negate the compactness of the support for initial distributions that are both smooth and possess compact support. The presented analysis in this work seeks to delve into the nature of solutions within open finite domains, with an ambition to scale up the findings to the realm of through the use of an extension operator. The foundation for establishing the existence and uniqueness of these solutions leans on the semigroup representation theory. Asymptotic solutions have been carved out via the Hamilton–Jacobi transport equation. To cement the theoretical foundation, numerical simulations have been incorporated shedding light on the accuracy of our initial assumptions within the asymptotic framework.

On non-regular semigroups: $\lambda$-semidirect products, Zappa-Szép products and representations

This article encompasses the study of non-regular semigroups. First, we show that the $\lambda$-semidirect product $M \rtimes^{\lambda} W$ of a left $P$-Ehresmann semigroup $M$ and a left restriction semigroup $W$ is a left $P$-Ehresmann semigroup. We explore the behavior of generalized Green's relations on $M \rtimes^{\lambda} W$, and investigate some properties of $M \rtimes^{\lambda} W$. Second, the Zappa-Szép product of a right Ehresmann semigroup and its distinguished semilattice is studied. Lastly, the theory of representations of left Ehresmann semigroups with zero via homomorphisms of left Ehresmann semigroups with zero into Clifford restriction semigroups with zero is presented.

On the minimal faithful degree of Rhodes semisimple semigroups

In this paper we compute the minimal degree of a faithful representation by partial transformations of a finite semigroup admitting a faithful completely reducible matrix representation over the field of complex numbers. This includes all inverse semigroups, and hence our results generalize earlier results of Easdown and Schein on the minimal faithful degree of an inverse semigroup. It also includes well-studied monoids like full matrix monoids over finite fields and the monoid of binary relations (i.e., matrices over the Boolean semiring). Our answer reduces the computation to considerations of permutation representations of maximal subgroups that are faithful when restricted to distinguished normal subgroups. This is analogous to (and inspired by) recent results of the second author on the minimal number of irreducible constituents in a faithful completely reducible complex matrix representation of a finite semigroup that admits one. To illustrate what happens when a finite semigroup does not admit a faithful completely reducible representation, we compute the minimal faithful degree of the opposite monoid of the full transformation monoid.

Congruence-simplicity of Steinberg algebras of non-Hausdorff ample groupoids over semifields

We investigate the algebra of an ample groupoid, introduced by Steinberg, over a semifield S . In particular, we obtain a complete characterization of congruence-simpleness for Steinberg algebras of second-countable ample groupoids, extending the well-known characterizations when S is a field. We apply our congruence-simplicity results to tight groupoids of inverse semigroup representations associated to self-similar graphs.

Which bath Hamiltonians matter for thermal operations?

In this article, we explore the set of thermal operations from a mathematical and topological point of view. First, we introduce the concept of Hamiltonians with a resonant spectrum with respect to some reference Hamiltonian, followed by proving that when defining thermal operations, it suffices to only consider bath Hamiltonians, which satisfy this resonance property. Next, we investigate the continuity of the set of thermal operations in certain parameters, such as energies of the system and temperature of the bath. We will see that the set of thermal operations changes discontinuously with respect to the Hausdorff metric at any Hamiltonian, which has the so-called degenerate Bohr spectrum, regardless of the temperature. Finally, we find a semigroup representation of (enhanced) thermal operations in two dimensions by characterizing any such operation via three real parameters, thus allowing for a visualization of this set. Using this, in the qubit case, we show commutativity of (enhanced) thermal operations and convexity of thermal operations without the closure. The latter is done by specifying the elements of this set exactly.

A canonical representation of maximal, indecomposable semi groups of constant-rank nonnegative r-potent matrices in ݑ0ݑ›(R)

An ݑŸ-potent matrix in ݑ0ݑ›(R) is an ݑ› × ݑ› matrix satisfying ݐذݑŸ = ݐ¸. A multiplicative semigroup S in ݑ0ݑ›(R) is said to be decomposable if there exists a special kind of common invariant subspace called standard invariant subspace for each ݐ´ ∈ S. A semi-group S of non-negative ݑŸ-potent matrices in ݑ0ݑ›(R) is known to be decomposable if rank (ݑ†) > ݑŸ − 1 for all ݑ† in S. Further, a semigroup S in ݑ0ݑ›(R) of nonnegative matrices will be called a full semigroup if S has no common zero row and no common zero column. We have studied the structure of maximal semi-groups of non-negative ݑŸ-potent matrices in ݑ0ݑ›(R) under the special condition of fullness. The objectives of this paper are twofold: (1) To find conditions under which semigroups of nonnegative r-potent matrices can be expressed as a direct sum of maximal rank-one indecomposable semigroups of r-potent matrices; and (2) To obtain a canonical representation of maximal indecomposable rank-one semigroups of r-potent matrices which in the light of the above result gives a complete characterization of such semigroups having constant rank.

Representations of Munn algebras and related semigroups

We establish representation types (finite, tame or wild) of finite dimensional Munn algebras with semisimple bases. As an application, we establish representation types of finite Rees matrix semigroups, in particular, 0-simple semigroups, and their mutually annihilating unions.

On the Representation of Semigroup C*-Algebra as a Crossed Product

On the Representation of Semigroup C*-Algebra as a Crossed Product

Σ-functions of categories of matrix representations of nilpotent semigroups

Σ-functions of categories of matrix representations of nilpotent semigroups

Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise

We provide a complete elaboration of the L^2-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behavior of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker–Planck framework and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and Röckner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.

Irreducible representations of Hecke–Kiselman monoids

Let K[HKΘ] denote the Hecke–Kiselman algebra of a finite oriented graph Θ over an algebraically closed field K. All irreducible representations, and the corresponding maximal ideals of K[HKΘ], are characterized in case this algebra satisfies a polynomial identity. The latter condition corresponds to a simple condition that can be expressed in terms of the graph Θ. The result shows a surprising similarity to the classical results on representations of finite semigroups; namely every representation either comes form an idempotent in the Hecke–Kiselman monoid HKΘ (and hence it is 1-dimensional), or it comes from certain semigroup of matrix type. The case when Θ is an oriented cycle plays a crucial role; the prime spectrum of K[HKΘ] is completely characterized in this case.

Uniform convergence of operator semigroups without time regularity

When we are interested in the long-term behaviour of solutions to linear evolution equations, a large variety of techniques from the theory of C_0-semigroups is at our disposal. However, if we consider for instance parabolic equations with unbounded coefficients on {mathbb {R}}^d, the solution semigroup will not be strongly continuous, in general. For such semigroups many tools that can be used to investigate the asymptotic behaviour of C_0-semigroups are not available anymore and, hence, much less is known about their long-time behaviour. Motivated by this observation, we prove new characterisations of the operator norm convergence of general semigroup representations—without any time regularity assumptions—by adapting the concept of the “semigroup at infinity”, recently introduced by M. Haase and the second named author. Besides its independence of time regularity, our approach also allows us to treat the discrete-time case (i.e. powers of a single operator) and even more abstract semigroup representations within the same unified setting. As an application of our results, we prove a convergence theorem for solutions to systems of parabolic equations with the aforementioned properties.

K-Theory for Semigroup C*-Algebras and Partial Crossed Products

Using the Baum–Connes conjecture with coefficients, we develop a K-theory formula for reduced C*-algebras of strongly 0-E-unitary inverse semigroups, or equivalently, for a class of reduced partial crossed products. This generalizes and gives a new proof of previous K-theory results of Cuntz, Echterhoff and the author. Our K-theory formula applies to a rich class of C*-algebras which are generated by partial isometries. For instance, as new applications which could not be treated using previous results, we discuss semigroup C*-algebras of Artin monoids, Baumslag-Solitar monoids and one-relator monoids, as well as C*-algebras generated by right regular representations of semigroups of number-theoretic origin, and C*-algebras attached to tilings.

The sympathetic sceptic’s guide to semigroup representations

This is an elementary introduction to the representation theory of finite semigroups. We illustrate the Clifford–Munn correspondence between the representations of a semigroup and the representations of its maximal subgroups. The emphasis throughout is on naturally occurring examples.

Amenability properties and fixed point properties of affine representations of semigroups

Amenability properties and fixed point properties of affine representations of semigroups

Semi-inner products and operators which attain their norm

It has been proved by Koehler and Rosenthal [Studia Math. 36 (1970), 213–216] that an linear isometry $U\in\mathcal{L}(X)$ preserves some semi-inner-product. Recently, similar investigations have been carried out by Niemiec and Wójcik for continuous representations of amenable semigroups into $\mathcal{L}(X)$ (cf. [Studia Math. 252 (2020), 27–48]). In this paper we generalize the result of Koehler and Rosenthal. Namely, we prove that if an operator $T\in \mathcal{L}(X)$ of norm one attains its norm then there is a semi-inner-product $[\cdot|\diamond]: X\times X\to \mathbf{F}$ that the operator $T$ preserves this semi-inner-product on the norm attaining set. More precisely, we show that the equality $[T(\cdot)|\mathit{Tx}]=[\cdot|x]$ holds for all $x\in M_{T}:=\{y\in S_{X}: \|\mathit{Ty}\|=1\}$.

Scalarly weakly almost periodic distal representations are strongly almost periodic

Kadets and Lyubich have shown that a weakly almost periodic representation of a topological group on a Banach space has the property that a vector with almost periodic coefficients is almost periodic. In this paper we obtain the analogous result for distal representations of semigroups on quasi-complete locally convex spaces, removing the condition that the representation be weakly almost periodic and requiring only that the coefficients of the representation be weakly almost periodic.

Markov chains under nonlinear expectation

AbstractIn this paper, we consider continuous‐time Markov chains with a finite state space under nonlinear expectations. We define so‐called Q‐operators as an extension of Q‐matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q‐operators in terms of a positive maximum principle, a dual representation by means of Q‐matrices, time‐homogeneous Markov chains under convex expectations, and a class of nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to the case of model uncertainty in the generator. We further derive an explicit primal and dual representation of convex semigroups arising from Markov chains under convex expectations via the Fenchel–Legendre transformation of the generator. We illustrate the results with several numerical examples, where we compute price bounds for European contingent claims under model uncertainty in terms of the rate matrix.

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Matrix representations of finite semigroups over fields are studied not so well as for finite groups. Representations of finite groups over fields are studied sufficiently well; in particular, the criterions of representation type are fully defined for an arbitrary field. If the characteristic $p$ of a field $K$ does not divide the order of a group (classical case), then the group has, up to equivalence, finite number of non-decomposable representations; such group is called a group of finite representation type. If the characteristic $p$ divides the order of a group (modular case), then the group has finite representation type only if its Sylow $p$-subgroup is cyclic. In this case for most finite groups the problems of describing their representations includes the problem on classification, up to similarity, of the pairs of matrices. Such groups are called wild, and groups that allow explicit descriptions of representations are called tame. The tame and wild groups in modular case are fully described by the first author together with Yu. A. Drozd. In the theory of representations of semigroups, the largest number of works are devoted to irreducible representations. Among the old results, there are only some results on semigroups of finite representation type, namely, for a finite quite simple semigroup (I. S. Ponizovsky) and some semigroups of all transformations of a finite set (I. S. Ponizovsky, C. Ringel). In cases, when the numbers of non-decomposable representations is infinite, the most famous are the results from the theory of representations of algebras that can be reformulated in terms of representations of semigroups: the description of representations of the algebra $ $ (I. M. Gelfand, V. A. Ponomarev and L. O. Nazarova, A. V. Roiter, V. V. Sergeichuk, V. M. Bondarenko) and the algebra $ $ (V. M. Bondarenko and C. Ringel). If we are not talking about individual semigroups, but about semigroup classes, then it should be noted works about on representations of the semigroups generated by idempotents with partial zero multiplication (V. M. Bondarenko, O. M. Tertychna), representations of the Rees semigroups (S. M. Dyachenko), semigroups generated by the potential elements (V. M. Bondarenko, O. V. Zubaruk) and representations of direct products of the symmetric second-order semigroup (V. M. Bondarenko, E. M. Kostyshyn). Such semigroups can have both a finite and infinite representation type. V. M. Bondarenko and Ja. V. Zatsikha described representation types of the third-order semigroups over a field, and indicate the canonical form of the matrix representations for any semigroup of finite representation type. This article is devoted to the study of similar problems for oversemigroups of commutative semigroups.

Inductive Sequences of Toeplitz Algebras and Limit Automorphisms

The note is concerned with inductive sequences of Toeplitz algebras. The Toeplitz algebra is the $$C^{*}$$ -subalgebra in the algebra of all bounded linear operators. This subalgebra is generated by the right shift operator on the Hilbert space of all square summable complex-valued functions defined on the additive semigroup of non-negative integers. We study the inductive sequences of Toeplitz algebras whose bonding $$\ast$$ -homomorphisms are defined by arbitrary sequences of natural numbers. The inductive limits of such sequences are the reduced semigroup $$C^{*}$$ -algebras generated by representations for semigroups of non-negative rational numbers. We consider the limit $$\ast$$ -endomorphisms of these inductive limits. Such an endomorphism is induced by a morphism between two copies of the same inductive sequence of Toeplitz algebras. We give the necessary and sufficient conditions for these endomorphisms to be $$\ast$$ -automorphisms of $$C^{*}$$ -algebras. These criteria are formulated in algebraic, number-theoretical and functional terms.