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Let G be a locally compact Hausdorff group, and let P(G) denote the class of all regular probability measures on G. It is well known that P(G) forms a semigroup under the convolution of measures. In this paper, we prove that P(G) is not algebraically regular in the sense that not every element has a generalized inverse. Additionally, we attempt to identify algebraically regular elements in some exceptional cases. Several supporting examples are provided to justify these assumptions. [ABSTRACT FROM AUTHOR]

Deriving from the motion of a three-dimensional compressible viscous heat-conducting fluid in an infinite layer bounded above by a free surface, the paper is concerned with the asymptotic behavior of solutions to the linearized compressible Navier-Stokes equations in a strip domain. With the help of the explicit solution formula for the corresponding resolvent problem we have achieved, the time-decay estimates of solutions to this problem in $ L^p $ spaces, $ 2\le p<\infty $, are well established. Moreover, since we are interested in the linearized dynamic boundary condition of the surface wave problem, some exponential decay properties of the solutions are revealed. Our result is crucial and indispensable for developing the $ L^p $ theory for the free boundary problem of the full compressible Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

Let S be a semigroup. In the present paper, we determine the complex-valued solutions (f, g) of the functional equation g(xσ(y)) = g(x)g(y) − f(x)f(y) + αf(xσ(y)), x, y ∈ S, where σ : S → S is an automorphism that need not be involutive, and α ∈ C is a fixed constant. Our results generalize and extend the ones by Stetkær in The cosine addition law with an additional term. Aequat Math., no. 6, 90, 1147-1168 (2016), and also the ones by Aserrar and Elqorachi in A generalization of the cosine addition law on semigroups. Aequat Math. 97, 787–804 (2023). Some consequences of our results are presented. [ABSTRACT FROM AUTHOR]

In the theory of nonadditive integrals, an indispensable step is to define a pair of pseudo-addition and pseudo-multiplication that fulfill the conditional distributivity, leading to a structure of an ordered semiring in some sense. In this paper, we focus on conditionally distributive uninorms locally internal on the boundary, show that the second involved uninorm must be locally internal, and present a general framework of structures of such a pair of uninorms. [ABSTRACT FROM AUTHOR]

Let S be a semigroup, Z(S) the center of S. In this paper, we determine the complex-valued solutions of Kannappan–d'Alembert's functional equation ∫ S f (x y t) d μ (t) + ∫ S f (σ (y) x t) d μ (t) = 2 f (y) f (x) , x , y ∈ S , and Kannappan–Wilson's functional equation ∫ S f (x y t) d μ (t) + ∫ S f (σ (y) x t) d μ (t) = 2 f (y) g (x) , x , y ∈ S , where μ is a measure that is a linear combination of Dirac measures (δ z i) i ∈ I , such that z i ∈ Z (S) for all i ∈ I , and σ : S → S is an involutive automorphism or an involutive anti-automorphism for the first equation and an involutive automorphism for the second one. We also give some interesting applications. [ABSTRACT FROM AUTHOR]

An algebra is finitely related (or has finite degree) if its term functions are determined by some finite set of finitary relations. Nilpotent monoids built from words, via Rees quotients of free monoids, have been used to exhibit many interesting properties with respect to the finite basis problem. We show that much of this intriguing behaviour extends to the world of finite relatedness by using interlocking patterns called chain, crown, and maelstrom words. In particular, we show that there are large classes of non-finitely related nilpotent monoids that can be used to construct examples of: ascending chains of varieties alternating between finitely and non-finitely related; non-finitely related semigroups whose direct product are finitely related; the addition of an identity element to a non-finitely related semigroup to produce a finitely related semigroup. [ABSTRACT FROM AUTHOR]

We develop the idempotent theory for algebras over a class of semigroups called left regular bands of groups (LRBGs), which simultaneously generalize group algebras of finite groups and left regular band (LRB) algebras. Our techniques weave together the representation theory of finite groups and LRBs, opening the door for a systematic study of LRBGs in an analogous way to LRBs. We apply our results to construct complete systems of primitive orthogonal idempotents in the Mantaci–Reutenauer algebra MR n [ G ] associated to any finite group G. When G is abelian, we give closed form expressions for these idempotents, and when G is the cyclic group of order two, we prove that they recover idempotents introduced by Vazirani. [ABSTRACT FROM AUTHOR]

Let M (S ; Λ ; P) denote a Rees I × Λ matrix semigroup over a semigroup S in which I is a singleton. In this paper we describe the right regular representation of M (S ; Λ ; P) , and find connection between properties of S and M (S ; Λ ; P) . We also prove embedding theorems on Rees matrix semigroups M (S ; Λ ; P) . [ABSTRACT FROM AUTHOR]

AbstractOur aim in this paper is to initiate the study of exponent semigroups for rational matrices. We prove that every numerical semigroup is the exponent semigroup of some rational matrix. We also obtain lower bounds on the size of such matrices and discuss the related class of power-integral matrices. [ABSTRACT FROM AUTHOR]

There is a long-standing problem, posed by A.T.-M. Lau [ Fixed point theory and its applications , Academic Press, New York-London, 1976, pp. 121–129], whether left amenability is sufficient to ensure the existence of a common fixed point for every jointly weak^{\ast } continuous nonexpansive semigroup action on a nonempty weak^{\ast } compact convex set in a dual Banach space. In this note we discuss the current status of this problem and give a partial solution in the case of weak^{\ast } compact convex sets with the Radon–Nikodým property. [ABSTRACT FROM AUTHOR]

This paper addresses the null controllability of an abstract boundary control systems in Hilbert spaces where the system operator is of Riesz type. Consequently, this document establishes a criterion for null controllability in such systems based on initial data, utilizing the moment problem. Furthermore, this criterion is formulated by employing a null controllability criterion that is applicable to a corresponding linear system with internal control. Finally, we apply our approach to the heat equation and the Mullins equation, demonstrating the practicality of our methodology. [ABSTRACT FROM AUTHOR]

Let S be a semigroup. Our main results are that we describe the complex-valued solutions of the following functional equations g (x σ (y)) = g (x) g (y) + f (x) f (y) , x , y ∈ S , f (x σ (y)) = f (x) g (y) + f (y) g (x) , x , y ∈ S , and f (x σ (y)) = f (x) g (y) - f (y) g (x) , x , y ∈ S , where σ : S → S is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants. We also give some applications. [ABSTRACT FROM AUTHOR]

In this paper, we partition, enumerate and classify hyper nearrings up to isomorphism of order less than 4 and then determine their automorphism groups. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the existence of strict solutions to a class of differential equations with state-dependent nonlocal conditions. Using fixed point theorems and general Gronwall inequality, we give some new conclusions. Moreover, an example is given to illustrate our conclusions valuable. [ABSTRACT FROM AUTHOR]

A significant development in the field of gyrogroups was the introduction of the space of all relativistically admissible velocities, which brought gyrogroups into the mainstream. A group has various generalizations, one of which is the notion of gyrogroups. Moreover, for any pair (a; b) in this structure, there exists an automorphism gyr[a; b] that fulfills left associativity and left loop property. The motivation behind this study is to generalize gyrogroups and semigroups, which has led to the introduction of gyrosemigroups. Accordingly, in this paper, some classes of gyrosemigroups are presented. Also, all gyrosemigroups of order 2 are characterized. Furthermore, the gyrosemigroups with an identity or a zero are studied. [ABSTRACT FROM AUTHOR]

Given a semigroup S , for each Green's relation K ∈ { L , R , J , H } on S , the K -height of S , denoted by H K (S) , is the height of the poset of K -classes of S. More precisely, if there is a finite bound on the sizes of chains of K -classes of S , then H K (S) is defined as the maximum size of such a chain; otherwise, we say that S has infinite K -height. We discuss the relationships between these four K -heights. The main results concern the class of stable semigroups, which includes all finite semigroups. In particular, we prove that a stable semigroup has finite L -height if and only if it has finite R -height if and only if it has finite J -height. In fact, for a stable semigroup S , if H L (S) = n then H R (S) ≤ 2 n − 1 and H J (S) ≤ 2 n − 1 , and we exhibit a family of examples to prove that these bounds are sharp. Furthermore, we prove that if 2 ≤ H L (S) < ∞ and 2 ≤ H R (S) < ∞ , then H J (S) ≤ H L (S) + H R (S) − 2. We also show that for each n ∈ N there exists a semigroup S such that H L (S) = H R (S) = 2 n + n − 3 and H J (S) = 2 n + 1 − 4. By way of contrast, we prove that for a regular semigroup the L -, R - and H -heights coincide with each other, and are greater or equal to the J -height. Moreover, in a stable, regular semigroup the L -, R -, H - and J -heights are all equal. [ABSTRACT FROM AUTHOR]

In this present paper, we concern to investigate the existence and uniqueness of mild solutions for a new class of fractional evolution problems involving sectorial operators and nonlocal conditions by means of fixed point theory. Here we use Caputo fractional derivative of order α with 2 < α < 3. In this sense, in order to elucidate the results obtained, we end the article with examples. [ABSTRACT FROM AUTHOR]

A set of binary relations closed with respect to some collection of operations on relations forms an algebra called an algebra of relations. The theory of algebras of relations is an essential part of modern algebraic logic and has numerous applications in semigroup theory. The following problems naturally arise when classes of algebras of relation are considered: find a system of axioms for these classes, and find a basis of of identities (quasi-identities) for the varieties (quasi-varieties) generated by these classes. In the paper, these problems are solved for the class of semigroups of relation with the binary associative operation of the rectangular product, the result of which is the Cartesian product of the first projection of the first relation on the second projection of the second one.

In this paper we study the complex symmetry in the several variable Fock space by using the techniques of weighted composition operators and semigroups. We characterize unbounded weighted composition operators that are (real) complex symmetric with respect to a concrete conjugation. Using this characterization, we study complex symmetric semigroups and their generators. We realize such generators as first-order differential operators. [ABSTRACT FROM AUTHOR]

Let S be a suitable subsemigroup of a locally compact abelian group and let T = {T (s)}s∈S be a bounded and strongly continuous representation of S on a Banach space X. In this note, we study the spectral conditions on T and the ergodic conditions on x ∈ X which will imply that T (s) x → 0 strongly through S. [ABSTRACT FROM AUTHOR]

Similar to how the quasi-interior ideal generalizes the ideal and interior ideal of a semigroup, the concept of soft intersection quasi-interior ideal generalizes the idea of soft intersection ideal and soft intersection interior ideal of a semigroup. In this study, we provide the notion of soft intersection almost quasi-interior ideal as well as the soft intersection weakly almost quasi-interior ideal in a semigroup. We show that any nonnull soft intersection quasi-interior ideal is a soft intersection almost quasi-interior ideal; and soft intersection almost quasi-interior ideal is a soft intersection weakly almost quasi-interior ideal, but the converses are not true. We further demonstrate that any idempotent soft intersection almost quasi-interior ideal is a soft intersection almost subsemigroup. With the established theorem that states that if a nonempty set A is almost quasi-interior ideal, then its soft characteristic function is a soft intersection almost quasi-interior ideal, and vice versa, we are also able to derive several intriguing relationships concerning minimality, primeness, semiprimeness, and strongly primeness between almost quasi-interior ideals, and soft intersection almost quasi-interior ideals. [ABSTRACT FROM AUTHOR]

This paper presents a spatiotemporal reaction–diffusion model for epidemics to predict how the infection spreads in a given space. The model is based on a system of partial differential equations with the Neumann boundary conditions. First, we study the existence and uniqueness of the solution of the model using the semigroup theory and demonstrate the boundedness of solutions. Further, the proposed model's basic reproduction number is calculated using the eigenvalue problem. Moreover, the dynamic behavior of the disease‐free steady states of the model for R0<1$$ {\mathcal{R}}_0<1 $$ is investigated. The uniform persistence of the model is also discussed. In addition, the global asymptotic stability of the endemic steady state is examined. Finally, the numerical simulations validate the theoretical results. [ABSTRACT FROM AUTHOR]

This study looks closely into the analysis of the variation of constants formula given by Φ (t) = S (t) Φ (0) + ∫ 0 t S (t − σ) F (σ , Φ (σ)) d σ , for t ∈ [ 0 , T ] , T > 0 , within the context of modular function spaces L ρ . Additionally, this research explores practical applications of the variation of constants formula in variable exponent Lebesgue spaces L p (·) . Specifically, the study examines these spaces under certain conditions applied to the exponent function p (·) and the functions F as well as the semigroup S (t) , utilizing the symmetry properties of the algebraic semigroup. This investigation sheds light on the intricate interplay between parameters and functions within these mathematical frameworks, offering valuable insights into their behavior and properties in L p (·) . [ABSTRACT FROM AUTHOR]

The objective of this paper is to study the existence, uniqueness and stability of solutions for some nonlocal impulsive evolution problems involving conformable fractional derivative. The main results are based on fixed point theorems combined with the semigroup theory. As applications, we give an example to demonstrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

We introduce the notion of a partially ordered abelian algebra which is a natural generalization of the notion of an abelian algebra. We describe partially ordered abelian groupoids with identity, finite quasigroups, regular semigroups, semisimple semigroups, and semigroups with minimal left and right ideals. [ABSTRACT FROM AUTHOR]

In this paper, the structure of the nearly invariant subspaces for discrete semigroups generated by several (even infinitely many) automorphisms of the unit disc is described. As part of this work, the near S ∗ -invariance property of the image space C φ (ker T) is explored for composition operators C φ , induced by inner functions φ , and Toeplitz operators T. After that, the analysis of nearly invariant subspaces for strongly continuous multiplication semigroups of isometries is developed with a study of cyclic subspaces generated by a single Hardy class function. These are characterised in terms of model spaces in all cases when the outer factor is a product of an invertible function and a rational (not necessarily invertible) function. Techniques used include the theory of Toeplitz kernels and reproducing kernels. [ABSTRACT FROM AUTHOR]

Through this work, we establish the periodicity of solutions for the following neutral differential equation defined as ddtGwt=LGwt+D(t,wt)+H(t)$$ \frac{d}{dt}\mathcal{G}{w}_t&#x0003D;\mathcal{LG}{w}_t&#x0002B;\mathcal{D}\left(t,{w}_t\right)&#x0002B;\mathcal{H}(t) $$. Applying the theory of perturbation of semi‐Fredholm operators, we purpose an a priori estimate on the bounded operator D$$ \mathcal{D} $$ to derive the periodicity of bounded solutions on [0,+∞)$$ \left[0,&#x0002B;\infty \right) $$ without taking of consideration the compactness of the semigroup generated by the part of L$$ \mathcal{L} $$. Moreover, we introduce an application includes numerical simulations to depict the applicability of theoretical findings. [ABSTRACT FROM AUTHOR]

Grillet (Semigroup Forum 50:25–36, 1995) introduced the concept of stratified semigroups as a kind of generalisation of finite nilsemigroups. We extend Grillet's ideas by introducing the notion of the base of a semigroup and show that a semigroup is stratified if and only if its base is either empty or consists of only the zero element. The general structure of semigroups with non-trivial bases is studied and we show that these can be described in terms of ideal extensions of semigroups by stratified semigroups. We consider certain types of group-bound semigroups and also ideal extensions of Clifford semigroups, and show how to describe them as semilattices of ideal extensions by stratified semigroups and provide a number of interesting examples. [ABSTRACT FROM AUTHOR]

In this paper we provide an account of the Todd–Coxeter algorithm for computing congruences on semigroups and monoids. We also give a novel description of an analogue for semigroups of the so-called Felsch strategy from the Todd–Coxeter algorithm for groups. [ABSTRACT FROM AUTHOR]

Let S be a semigroup, Z (S) the center of S and σ : S → S is an involutive automorphism. Our main results is that we describe the solutions of the Kannappan-Wilson functional equation ∫ S f (x y t) d μ (t) + ∫ S f (σ (y) x t) d μ (t) = 2 f (x) g (y) , x , y ∈ S , and the Van Vleck-Wilson functional equation ∫ S f (x y t) d μ (t) - ∫ S f (σ (y) x t) d μ (t) = 2 f (x) g (y) , x , y ∈ S , where μ is a measure that is a linear combination of Dirac measures (δ z i) i ∈ I , such that z i ∈ Z (S) for all i ∈ I . Interesting consequences of these results are presented. [ABSTRACT FROM AUTHOR]

Let S be a semigroup and K a field. A function f : S → K is additive if f (x y) = f (x) + f (y) for all x , y ∈ S , and functions g , h : S → K form a sine pair if they satisfy the sine addition law g (x y) = g (x) h (y) + h (x) g (y) for all x , y ∈ S . Adding these two equations we arrive at the functional equation (*) f (x y) + g (x y) = f (x) + f (y) + g (x) h (y) + h (x) g (y) . The alienation question for additivity and sine additivity asks whether (*) implies that f is additive and (g, h) is a sine pair. To fully answer this question we find the general solution of (*) for unknown functions f , g , h : S → C . The solution illustrates a significant amount of interdependence between additivity and sine additivity. [ABSTRACT FROM AUTHOR]

It is shown that any ring being a sum of two left T-nilpotent subrings is left T-nilpotent. The paper contains the description of all the semigroups S such that an S-graded ring R = ⨁ s ∈ S A s has the property that the left T-nilpotency of all subrings among the subgroups A s of the additive group of R implies the left T-nilpotency of R. Furthermore, this result is extended to rings R being S-sums. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to revisit previous works of the author with Helffer and Sjöstrand (arXiv:1001.4171v1. 2010; Int Equ Op Theory 93(3):36, 2021) on the stability of semigroups which were proved in the Hilbert case by considering the Banach case at the light of a paper by Latushkin and Yurov (Discrete Contin Dyn Syst 33:5203–5216, 2013). [ABSTRACT FROM AUTHOR]

This study mainly concerns the controllability of semilinear noninstantaneous impulsive neutral stochastic differential equations via the Atangana-Baleanu (AB) Caputo fractional derivative (FD). The essential findings are created using methods and concepts from semigroup theory, stochastic theory, fractional calculus, K-set contraction, and measure of noncompactness. Finally, an example is provided to demonstrate the applications of the key findings.

This article has two main parts. In the first part we show that some of the basic theory of generalized polynomials on commutative semi-groups can be extended to all semigroups. In the second part we show that if a sub-semigroup S of a group G generates G in the sense that G = S · S−1, then a generalized polynomial on S with values in an Abelian group H can be extended to a generalized polynomial on G into H. Finally there is a short discussion of the extendability of exponential functions and generalized exponential polynomials.

In this paper, we prove new results which make it possible to vastly simplify solution of the problem of Nathanson concerning the enlargement of support bases and thus of the corresponding semigroup supports of rational function solutions to the quantum functional equations arising from multiplication of quantum integers. In particular, we establish a certain quantum equivalence relation on the set of all prime numbers which encodes the symmetry of these solutions and thus solve a problem which simplify in crucial ways the criteria for the problem posed by Nathanson concerning extensions of the support bases of these solutions.

We investigate a non-conservative semigroup $ (S_{t})_{t\ge 0} $ (S t ) t ≥ 0 determined by a branching process tracing the evolution of particles moving in a domain in $ {\mathbb R}^{d} $ R d . When a particle is killed at the boundary, a new generation of particles with mean number $ \bar {K} $ K ¯ is born at a random point in the domain. Between branching, the particles are driven by a diffusion process with Dirichlet boundary conditions. According to the sign of $ \bar {K}-1 $ K ¯ − 1 , we distinguish super/sub-critical regimes and determine the exact exponential rate for the total number of particles $ n(t)\sim \exp (\alpha ^{*}t) $ n (t) ∼ exp ⁡ (α ∗ t) , with $ \alpha ^{*} $ α ∗ depending explicitly on $ \bar {K} $ K ¯ . We prove the Yaglom limit $ S_{t}/n(t)\to \nu $ S t / n (t) → ν , where the quasi-stationary distribution ν is determined by the resolvent of the Dirichlet kernel at the point $ \alpha ^{*} $ α ∗ . The main application is in particle systems, where the normalization of the semigroup by its total mass gives the hydrodynamic limit of the Bak-Sneppen branching diffusions (BSBD). Since ν is the asymptotic profile under equilibrium, and the family of quasi-stationary distributions ν is indexed by $ \bar {K} $ K ¯ , the model provides an explicit example of self-organized criticality. [ABSTRACT FROM AUTHOR]

This paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic semigroup. The main difficulty with these equations is how to define the concept of mild solutions due to the chosen type of unbounded perturbations. To overcome this problem, we first proved a regularity property of the stochastic convolution with respect to the domain of 'admissible' unbounded linear operators (not necessarily closed or closable). This is done using Yosida extensions of such unbounded linear operators. After proving the well-posedness of these equations, we also establish the Feller property for the corresponding transition semigroups. Several examples like heat equations and Schrödinger equations with nonlocal perturbations terms are given. Finally, we give an application to a general class of semilinear neutral stochastic equations. [ABSTRACT FROM AUTHOR]

This study mainly concerns the controllability of semilinear noninstantaneous impulsive neutral stochastic differential equations via the Atangana-Baleanu (AB) Caputo fractional derivative (FD). The essential findings are created using methods and concepts from semigroup theory, stochastic theory, fractional calculus, K -set contraction, and measure of noncompactness. Finally, an example is provided to demonstrate the applications of the key findings. [ABSTRACT FROM AUTHOR]

The theories of rough sets (RSs) and soft sets (SSs) are practical mathematical techniques to accommodate data uncertainty. On the other hand, fuzzy bipolar soft sets (FBSSs) can address uncertainty and bipolarity in various situations. The key objective of this study is to establish the notions of rough fuzzy bipolar soft ideals in semigroup (SG), which is an extension of the idea of rough fuzzy bipolar soft sets in a SG. Also, we have analyzed the roughness in the bipolar fuzzy subsemigroup (BF-SSG) by employing a congruence relation (CR) defined on the SG and investigating several related characteristics. Further, the idea is expanded to the rough fuzzy bipolar soft ideal, rough fuzzy bipolar soft interior ideal, and rough fuzzy bipolar soft bi-ideal in SGs. Moreover, it is observed that CR and complete CR (CCR) are critical in developing rough approximations of fuzzy bipolar soft ideals. Therefore, their related characteristics are studied via CRs and CCRs. [ABSTRACT FROM AUTHOR]

Let S be a semigroup, and σ: S → S an automorphism that need not be involutive. We determine the complex-valued solutions of the following functional equation f(xy) - µ(y) f(σ(y)x) = 2f(x)g(y), x,y ∈ S, where µ: S→C is a multiplicative function such that µ(xσ(x)) = 1 for all x ∈ S. This enables us to solve the functional equation f (xφ(y))-f(ψ(y)x) = 2f(x)g(y), x,y ∈ S, where φ, ψ: S → S are automorphisms such that φ is involutive and ψ is not necessarily involutive. Some consequences of these results are presented. [ABSTRACT FROM AUTHOR]

A family U of non-empty subsets of a set D is called an upfamily if for each set U E U any set F U U belongs to U. The upfamily extension υ(D) of D consists of all upfamilies on D. Any associative binary operation ∗: D×D → D can be extended to an associative binary operation ... In the paper, we show that the upfamily extension (υ(D), -|, -|) of a (strong) doppelsemigroup (D, -|, -|) is a (strong) doppelsemigroup as well and study some properties of this extension. Also we introduce the upfamily functor in the category DSG whose objects are doppelsemigroups and morphisms are doppelsemigroup homomorphisms. We prove that the automorphism group of the upfamily extension of a doppelsemigroup (D, -|, -|) of cardinality |D| ≥ 2 contains a subgroup, isomorphic to C2 × Aut(D, -|, -|). Also we describe the structure of upfamily extensions of all two-element doppelsemigroups and their automorphism groups. [ABSTRACT FROM AUTHOR]

The lattice cohomology and the graded root of an isolated curve singularity were recently introduced in [3]. The lattice cohomology is a categorification of the d-invariant. The hope is that for plane curve singularities it encodes subtle information about the analytic structure and concrete analytic invariants. The present paper is a positive result in this direction: we prove that the multiplicity of an irreducible plane curve singularity can be recovered from its lattice cohomology (or, from its graded root). In fact, we give four distinct proofs of this statement, each of them emphasizing a rather different aspect of the theory of plane curve germs. With these proofs, we also create new bridges between the abstract analytic type and the embedded topological type of the germ. In particular, we provide a new characterization of the Ap'ery set of the semigroup of the germ in terms of embedded data. [ABSTRACT FROM AUTHOR]

In this work, we will submit the form of the solution of the generalization kind of unbounded operator equation define on Hilbert space which is X*B*+BX = C, where X, is the bounded operator satisfy the above operator equations and showing this solution via application the semigroup theory, moreover we will discuss some properties of this solution and proved its unique. Also, it was proved that for a locally bounded operator X(n) satisfying the equation. [ABSTRACT FROM AUTHOR]