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16 pages, 278 KiB

Open AccessArticle

Zappa–Szép Groupoids of Inverse Semigroups and an Alternative Proof of Billhardt’s λ-Semidirect Products

by Suha Wazzan

Mathematics 2025, 13(7), 1122; https://doi.org/10.3390/math13071122 - 28 Mar 2025

Viewed by 194

The aim of this paper is to introduce and study Zappa-Szép groupoids of inverse semigroups. Some properties of such kinds of groupoids are explored. As an application, an alternative proof of Billhardt’s

λ

-semidirect products is given. We finish with several examples that [...] Read more.

The aim of this paper is to introduce and study Zappa-Szép groupoids of inverse semigroups. Some properties of such kinds of groupoids are explored. As an application, an alternative proof of Billhardt’s

λ

-semidirect products is given. We finish with several examples that highlight the versatility and applicability of Zappa-Szép groupoids in various types of inverse semigroups. Full article

(This article belongs to the Special Issue Theory and Application of Algebraic Combinatorics)

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27 pages, 392 KiB

Open AccessArticle

L1 Scheme for Semilinear Stochastic Subdiffusion with Integrated Fractional Gaussian Noise

by Xiaolei Wu and Yubin Yan

Fractal Fract. 2025, 9(3), 173; https://doi.org/10.3390/fractalfract9030173 - 12 Mar 2025

Viewed by 637

Abstract

This paper considers a numerical method for solving the stochastic semilinear subdiffusion equation which is driven by integrated fractional Gaussian noise and the Hurst parameter

H \in (1 / 2, 1)

. The finite element method is employed for spatial [...] Read more.

This paper considers a numerical method for solving the stochastic semilinear subdiffusion equation which is driven by integrated fractional Gaussian noise and the Hurst parameter

H \in (1 / 2, 1)

. The finite element method is employed for spatial discretization, while the L1 scheme and Lubich’s first-order convolution quadrature formula are used to approximate the Caputo time-fractional derivative of order

α \in (0, 1)

and the Riemann–Liouville time-fractional integral of order

γ \in (0, 1)

, respectively. Using the semigroup approach, we establish the temporal and spatial regularity of the mild solution to the problem. The fully discrete solution is expressed as a convolution of a piecewise constant function with the inverse Laplace transform of a resolvent-related function. Based on the Laplace transform method and resolvent estimates, we prove that the proposed numerical scheme has the optimal convergence order

O (τ^{min {H + α + γ - 1 - ε, α}}), ε > 0

. Numerical experiments are presented to validate these theoretical convergence orders and demonstrate the effectiveness of this method. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

16 pages, 320 KiB

Open AccessArticle

Idempotent-Aided Factorizations of Regular Elements of a Semigroup

by Miroslav Ćirić, Jelena Ignjatović and Predrag S. Stanimirović

Mathematics 2024, 12(19), 3136; https://doi.org/10.3390/math12193136 - 7 Oct 2024

Cited by 1 | Viewed by 933

Abstract

In the present paper, we introduce the concept of idempotent-aided factorization (I.-A. factorization) of a regular element of a semigroup, which can be understood as a semigroup-theoretical extension of full-rank factorization of matrices over a field. I.-A. factorization of a regular element d [...] Read more.

In the present paper, we introduce the concept of idempotent-aided factorization (I.-A. factorization) of a regular element of a semigroup, which can be understood as a semigroup-theoretical extension of full-rank factorization of matrices over a field. I.-A. factorization of a regular element d is defined by means of an idempotent e from its Green’s

D

-class as decomposition into the product

d = u v

, so that the element u belongs to the Green’s

R

-class of the element d and the Green’s

L

-class of the idempotent e, while the element v belongs to the Green’s

L

-class of the element d and the Green’s

R

-class of the idempotent e. The main result of the paper is a theorem which states that each regular element of a semigroup possesses an I.-A. factorization with respect to each idempotent from its Green’s

D

-class. In addition, we prove that when one of the factors is given, then the other factor is uniquely determined. I.-A. factorizations are then used to provide new existence conditions and characterizations of group inverses and

(b, c)

-inverses in a semigroup. In our further research, these factorizations will be applied to matrices with entries in a field, and efficient algorithms for realization of such factorizations will be provided. Full article

(This article belongs to the Section A: Algebra and Logic)

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13 pages, 297 KiB

Open AccessArticle

On Centralizers of Idempotents with Restricted Range

by Dilawar J. Mir and Amal S. Alali

Symmetry 2024, 16(6), 769; https://doi.org/10.3390/sym16060769 - 19 Jun 2024

Cited by 2 | Viewed by 1168

Abstract

This study delves into the structure and properties of left inverse zero divisor bands within semigroups, identifying their maximal forms and broadening the theoretical landscape of semigroup analysis. A significant focus is placed on the automorphisms of a semigroup S of centralizers of [...] Read more.

This study delves into the structure and properties of left inverse zero divisor bands within semigroups, identifying their maximal forms and broadening the theoretical landscape of semigroup analysis. A significant focus is placed on the automorphisms of a semigroup S of centralizers of idempotent transformations with restricted range, revealing that these automorphisms are inner ones and induced by the units of S. Additionally, we establish that the automorphism group

Aut (S)

is isomorphic to

U_{S}

, the group of units of S. These findings extend previous results on semigroups of transformations, enhancing their applicability and providing a more unified theory. The practical implications of this work span multiple fields, including automata theory, coding theory, cryptography, and graph theory, offering tools for more efficient algorithms and models. By simplifying complex concepts and providing a solid foundation for future research, this study makes significant contributions to both theoretical and applied mathematics. Full article

(This article belongs to the Special Issue Algebraic Systems, Models and Applications)

15 pages, 344 KiB

Open AccessArticle

Quasi-Semilattices on Networks

by Yanhui Wang and Dazhi Meng

Axioms 2023, 12(10), 943; https://doi.org/10.3390/axioms12100943 - 30 Sep 2023

Cited by 4 | Viewed by 1944

Abstract

This paper introduces a representation of subnetworks of a network

Γ

consisting of a set of vertices and a set of relations, where relations are the primitive structures of a network. It is proven that all connected subnetworks of a network

Γ

form [...] Read more.

This paper introduces a representation of subnetworks of a network

Γ

consisting of a set of vertices and a set of relations, where relations are the primitive structures of a network. It is proven that all connected subnetworks of a network

Γ

form a quasi-semilattice

L (Γ)

, namely a network quasi-semilattice.Two equivalences

σ

and

δ

are defined on

L (Γ)

. Each

δ

class forms a semilattice and also has an order structure with the maximum element and minimum elements. Here, the minimum elements correspond to spanning trees in graph theory. Finally, we show how graph inverse semigroups, Leavitt path algebras and Cuntz–Krieger graph

C^{*}

-algebras are constructed in terms of relations. Full article

(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)

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19 pages, 347 KiB

Open AccessArticle

On Novel Fractional Operators Involving the Multivariate Mittag–Leffler Function

by Muhammad Samraiz, Ahsan Mehmood, Saima Naheed, Gauhar Rahman, Artion Kashuri and Kamsing Nonlaopon

Mathematics 2022, 10(21), 3991; https://doi.org/10.3390/math10213991 - 27 Oct 2022

Cited by 13 | Viewed by 1547

Abstract

The multivariate Mittag–Leffler function is introduced and used to establish fractional calculus operators. It is shown that the fractional derivative and integral operators are bounded. Some fundamental characteristics of the new fractional operators, such as the semi-group and inverse characteristics, are studied. As [...] Read more.

The multivariate Mittag–Leffler function is introduced and used to establish fractional calculus operators. It is shown that the fractional derivative and integral operators are bounded. Some fundamental characteristics of the new fractional operators, such as the semi-group and inverse characteristics, are studied. As special cases of these novel fractional operators, several fractional operators that are already well known in the literature are acquired. The generalized Laplace transform of these operators is evaluated. By involving the explored fractional operators, a kinetic differintegral equation is introduced, and its solution is obtained by using the Laplace transform. As a real-life problem, a growth model is developed and its graph is sketched. Full article

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12 pages, 289 KiB

Open AccessArticle

Symmetric Properties of (b,c)-Inverses

by Guiqi Shi and Jianlong Chen

Mathematics 2022, 10(16), 2948; https://doi.org/10.3390/math10162948 - 16 Aug 2022

Cited by 2 | Viewed by 1389

Abstract

Let b and c be two elements in a semigroup S. The

(b, c)

-inverse is an important outer inverse because it unifies many common generalized inverses. This paper is devoted to presenting some symmetric properties of [...] Read more.

Let b and c be two elements in a semigroup S. The

(b, c)

-inverse is an important outer inverse because it unifies many common generalized inverses. This paper is devoted to presenting some symmetric properties of

(b, c)

-inverses and

(c, b)

-inverses. We first find that S contains a

(b, c)

-invertible element if and only if it contains a

(c, b)

-invertible element. Then, for four given elements

a, b, c, d

in S, we prove that a is

(b, c)

-invertible and d is

(c, b)

-invertible if and only if

a b d

is invertible along c and

d c a

is invertible along b. Inspired by this result, the

(b, c)

-invertibility is characterized by one-sided invertible elements. Furthermore, we show that a is inner

(b, c)

-invertible and d is inner

(c, b)

-invertible if and only if c is inner

(a, d)

-invertible and b is inner

(d, a)

-invertible. Full article

16 pages, 494 KiB

Open AccessArticle

Left (Right) Regular and Transposition Regular Semigroups and Their Structures

by Xiaohong Zhang and Yudan Du

Mathematics 2022, 10(7), 1021; https://doi.org/10.3390/math10071021 - 22 Mar 2022

Cited by 8 | Viewed by 3114

Abstract

Regular semigroups and their structures are the most wonderful part of semigroup theory, and the contents are very rich. In order to explore more regular semigroups, this paper extends the relevant classical conclusions from a new perspective: by transforming the positions of the [...] Read more.

Regular semigroups and their structures are the most wonderful part of semigroup theory, and the contents are very rich. In order to explore more regular semigroups, this paper extends the relevant classical conclusions from a new perspective: by transforming the positions of the elements in the regularity conditions, some new regularity conditions (collectively referred to as transposition regularity) are obtained, and the concepts of various transposition regular semigroups are introduced (L1/L2/L3, R1/R2/R3-transposition regular semigroups, etc.). Their relations with completely regular semigroups and left (right) regular semigroups, proposed by Clifford and Preston, are analyzed. Their properties and structures are studied from the aspects of idempotents, local identity elements, local inverse elements, subsemigroups and so on. Their decomposition theorems are proved respectively, and some new necessary and sufficient conditions for semigroups to become completely regular semigroups are obtained. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

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19 pages, 364 KiB

Open AccessArticle

Parameter–Elliptic Fourier Multipliers Systems and Generation of Analytic and C^∞ Semigroups

by Bienvenido Barraza Martínez, Jonathan González Ospino, Rogelio Grau Acuña and Jairo Hernández Monzón

Mathematics 2022, 10(5), 751; https://doi.org/10.3390/math10050751 - 26 Feb 2022

Viewed by 1593

Abstract

We consider Fourier multiplier systems on

R^{n}

with components belonging to the standard Hörmander class

S_{1, 0}^{m} (R^{n})

, but with limited regularity. Using a notion of parameter-ellipticity with respect to a subsector

Λ \subset C

(introduced by [...] Read more.

We consider Fourier multiplier systems on

R^{n}

with components belonging to the standard Hörmander class

S_{1, 0}^{m} (R^{n})

, but with limited regularity. Using a notion of parameter-ellipticity with respect to a subsector

Λ \subset C

(introduced by Denk, Saal, and Seiler) we show the generation of both

C^{\infty}

semigroups and analytic semigroups (in a particular case) on the Sobolev spaces

W_{p}^{k} (R^{n}, C^{q})

with

k \in N_{0}

,

1 \leq p < \infty

and

q \in N

. For the proofs, we modify and improve a crucial estimate from Denk, Saal and Seiler, on the inverse matrix of the symbol (see Lemma 2). As examples, we apply the theory to solve the heat equation, a linear thermoelastic plate equation, a structurally damped plate equation, and a generalized plate equation, all in the whole space, in the frame of Sobolev spaces. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

20 pages, 379 KiB

Open AccessArticle

Higher Regularity, Inverse and Polyadic Semigroups

by Steven Duplij

Universe 2021, 7(10), 379; https://doi.org/10.3390/universe7100379 - 13 Oct 2021

Viewed by 1877

Abstract

We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, [...] Read more.

We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions. Finally, we introduce the sandwich higher polyadic regularity with generalized idempotents. Full article

(This article belongs to the Special Issue Gauge Theory, Strings and Supergravity)

18 pages, 333 KiB

Open AccessArticle

Controllability of Semilinear Multi-Valued Differential Inclusions with Non-Instantaneous Impulses of Order α ∈ (1,2) without Compactness

by Zainab Alsheekhhussain and Ahmed Gamal Ibrahim

Symmetry 2021, 13(4), 566; https://doi.org/10.3390/sym13040566 - 29 Mar 2021

Cited by 3 | Viewed by 1589

Abstract

Herein, we investigated the controllability of a semilinear multi-valued differential equation with non-instantaneous impulses of order

α \in (1, 2)

, where the linear part is a strongly continuous cosine family without compactness. We did not assume any compactness conditions [...] Read more.

Herein, we investigated the controllability of a semilinear multi-valued differential equation with non-instantaneous impulses of order

α \in (1, 2)

, where the linear part is a strongly continuous cosine family without compactness. We did not assume any compactness conditions on either the semi-group, the multi-valued function, or the inverse of the controllability operator, which is different from the previous literature. Full article

12 pages, 298 KiB

Open AccessArticle

The Abelian Kernel of an Inverse Semigroup

by A. Ballester-Bolinches and V. Pérez-Calabuig

Mathematics 2020, 8(8), 1219; https://doi.org/10.3390/math8081219 - 24 Jul 2020

Cited by 1 | Viewed by 2091

Abstract

The problem of computing the abelian kernel of a finite semigroup was first solved by Delgado describing an algorithm that decides whether a given element of a finite semigroup S belongs to the abelian kernel. Steinberg extended the result for any variety of [...] Read more.

The problem of computing the abelian kernel of a finite semigroup was first solved by Delgado describing an algorithm that decides whether a given element of a finite semigroup S belongs to the abelian kernel. Steinberg extended the result for any variety of abelian groups with decidable membership. In this paper, we used a completely different approach to complete these results by giving an exact description of the abelian kernel of an inverse semigroup. An abelian group that gives this abelian kernel was also constructed. Full article

16 pages, 278 KiB

Open AccessArticle

Self-Similar Inverse Semigroups from Wieler Solenoids

by Inhyeop Yi

Mathematics 2020, 8(2), 266; https://doi.org/10.3390/math8020266 - 17 Feb 2020

Viewed by 1902

Abstract

Wieler showed that every irreducible Smale space with totally disconnected local stable sets is an inverse limit system, called a Wieler solenoid. We study self-similar inverse semigroups defined by s-resolving factor maps of Wieler solenoids. We show that the groupoids of germs [...] Read more.

Wieler showed that every irreducible Smale space with totally disconnected local stable sets is an inverse limit system, called a Wieler solenoid. We study self-similar inverse semigroups defined by s-resolving factor maps of Wieler solenoids. We show that the groupoids of germs and the tight groupoids of these inverse semigroups are equivalent to the unstable groupoids of Wieler solenoids. We also show that the

C^{*}

-algebras of the groupoids of germs have a unique tracial state. Full article

(This article belongs to the Special Issue Operator Algebras)

21 pages, 454 KiB

Open AccessArticle

Regular CA-Groupoids and Cyclic Associative Neutrosophic Extended Triplet Groupoids (CA-NET-Groupoids) with Green Relations

by Wangtao Yuan and Xiaohong Zhang

Mathematics 2020, 8(2), 204; https://doi.org/10.3390/math8020204 - 6 Feb 2020

Cited by 6 | Viewed by 2227

Abstract

Based on the theories of AG-groupoid, neutrosophic extended triplet (NET) and semigroup, the characteristics of regular cyclic associative groupoids (CA-groupoids) and cyclic associative neutrosophic extended triplet groupoids (CA-NET-groupoids) are further studied, and some important results are obtained. In particular, the following conclusions are [...] Read more.

Based on the theories of AG-groupoid, neutrosophic extended triplet (NET) and semigroup, the characteristics of regular cyclic associative groupoids (CA-groupoids) and cyclic associative neutrosophic extended triplet groupoids (CA-NET-groupoids) are further studied, and some important results are obtained. In particular, the following conclusions are strictly proved: (1) an algebraic system is a regular CA-groupoid if and only if it is a CA-NET-groupoid; (2) if (S, *) is a regular CA-groupoid, then every element of S lies in a subgroup of S, and every

ℋ

-class in S is a group; and (3) an algebraic system is an inverse CA-groupoid if and only if it is a regular CA-groupoid and its idempotent elements are commutative. Moreover, the Green relations of CA-groupoids are investigated, and some examples are presented for studying the structure of regular CA-groupoids. Full article

(This article belongs to the Special Issue New Challenges in Neutrosophic Theory and Applications)

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20 pages, 286 KiB

Open AccessArticle

Generalized Abel-Grassmann’s Neutrosophic Extended Triplet Loop

by Xiaogang An, Xiaohong Zhang and Yingcang Ma

Mathematics 2019, 7(12), 1206; https://doi.org/10.3390/math7121206 - 9 Dec 2019

Cited by 5 | Viewed by 1933

Abstract

A group is an algebraic system that characterizes symmetry. As a generalization of the concept of a group, semigroups and various non-associative groupoids can be considered as algebraic abstractions of generalized symmetry. In this paper, the notion of generalized Abel-Grassmann’s neutrosophic extended triplet [...] Read more.

A group is an algebraic system that characterizes symmetry. As a generalization of the concept of a group, semigroups and various non-associative groupoids can be considered as algebraic abstractions of generalized symmetry. In this paper, the notion of generalized Abel-Grassmann’s neutrosophic extended triplet loop (GAG-NET-Loop) is proposed and some properties are discussed. In particular, the following conclusions are strictly proved: (1) an algebraic system is an AG-NET-Loop if and only if it is a strong inverse AG-groupoid; (2) an algebraic system is a GAG-NET-Loop if and only if it is a quasi strong inverse AG-groupoid; (3) an algebraic system is a weak commutative GAG-NET-Loop if and only if it is a quasi Clifford AG-groupoid; and (4) a finite interlaced AG-(l,l)-Loop is a strong AG-(l,l)-Loop. Full article

(This article belongs to the Special Issue New Challenges in Neutrosophic Theory and Applications)

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