Topological graph inverse semigroups

Leavitt path algebras of labelled graphs

In this paper, we present results concerning the structure of the ideals in the Leavitt path algebra of a (countable) directed graph with coefficients in an integral domain, such as, describing the set of generators for an ideal; the necessary and sufficient conditions for an ideal to be prime; the necessary and sufficient conditions for a Leavitt path algebra to be simple. Besides, some other interesting properties of ideal structure in a Leavitt path algebra are also mentioned.

Let [Formula: see text] be the Leavitt path algebra of a directed graph [Formula: see text] over a field [Formula: see text]. In this paper, we determine [Formula: see text] and [Formula: see text] for the Lie algebra [Formula: see text] and the Jordan algebra [Formula: see text] arising from [Formula: see text] with respect to the standard involution to be solvable.

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be any directed graph, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> any field. We classify those graphs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the Leavitt path algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript upper K Baseline left-parenthesis upper E right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L_K(E)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is primitive. As a consequence, we obtain classes of examples of von Neumann regular prime rings which are not primitive.

Extensions of simple modules over Leavitt path algebras

This article investigates the fixed-time consensus control problem for nonlinear second-order multi-agent systems subjected to measurement noises and in absence of the second-order state measurements of the leader and the followers under directed communication topology graphs. For each follower, an innovative augmented fixed-time disagreement error observer (AFTDEO) is first designed to reconstruct the disagreement error within fixed time. On one hand, the proposed AFTDEO can not only operate well only depending on the noisy measurements of the first-order states of the leader and the followers, but also suppress the measurement noises of the leader and the followers. On the other hand, the proposed AFTDEO can guarantee that the observation errors are convergent to the origin in fixed time independent of the initial observation errors. Based on the proposed AFTDEO, a novel fixed-time consensus control scheme under a directed graph having a spanning tree is constructed to address the consensus problem without the second-order measurements of the leader and the followers by utilizing fixed-time nonsmooth backstepping technique. The fixed-time convergence of the tracking error is guaranteed through Lyapunov stability analysis. Finally, simulation results demonstrate the effectiveness of the proposed consensus control scheme.

On topological McAlister semigroups

In this chapter we investigate the ideal structure of Leavitt path algebras. We start by describing the natural \(\mathbb{Z}\)-grading on L K (E). We then present the Reduction Theorem; this result describes how elements of L K (E) may be transformed in some specified way to either a vertex or a cycle without exits. Numerous consequences are discussed, including the Uniqueness Theorems. We then establish in the Structure Theorem for Graded Ideals a precise relationship between graded ideals and explicit sets of idempotents (arising from hereditary and saturated subsets of vertices, together with breaking vertices). With this description of the graded ideals having been achieved, we focus in the remainder of the chapter on the structure of all ideals. We achieve in the Structure Theorem for Ideals an explicit description of the entire ideal structure of L K (E) (including both the graded and non-graded ideals) for an arbitrary graph E and field K. This result utilizes the Structure Theorem for Graded Ideals together with the analysis of the ideal generated by vertices which lie on cycles having no exits. A number of ring-theoretic results follow almost immediately from the Structure Theorem for Ideals, including the Simplicity Theorem. Along the way, we describe the socle of a Leavitt path algebra, and we achieve a description of the finite dimensional Leavitt path algebras.

In this paper we are concerned with continua that possess cut points and that admit the structure of topological semigroups, herein called mobs. First, we discuss the relations between the existence of cut points in compact connected mobs and the algebraic structure of the mob. Next, we consider D-chains, as defined by Remage [6], in compact connected mobs. D-chains may be considered to be generalizations of the classical A-sets. In the final section, we center our attention on compact metric trees, or dendrites, that are mobs and show that under certain circumstances the algebraic structure of the non-cut points has a marked influence upon the algebraic structure of the entire mob. The author wishes to acknowledge the advice and helpful suggestions of Professor A. D. Wallace in the preparation of this paper.

Universal quantum semigroupoids

The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures such as groups, rings, and fields. If the vertex set of a graph G^ is fully made up of the zero divisors of the modular ring Zn, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (modn), they are regarded as neighbors. Entropy, a notion taken from information theory and used in graph theory, measures the degree of uncertainty or unpredictability associated with a graph or its constituent elements. Entropy measurements may be used to calculate the structural complexity and information complexity of graphs. The first, second and second modified Zagrebs, general and inverse general Randics, third and fifth symmetric divisions, harmonic and inverse sum indices, and forgotten topological indices are a few topological indices that are examined in this article for particular families of zero-divisor graphs. A numerical and graphical comparison of computed topological indices over a proposed structure has been studied. Furthermore, different kinds of entropies, such as the first, second, and third redefined Zagreb, are also investigated for a number of families of zero-divisor graphs.

The multiplicative ideal theory of Leavitt path algebras

This article aims to provide a comprehensive investigation into the concept of ideals within various semigroup classes, focusing on their properties, characterization, and role in algebraic structures. The research will cover classical semigroup theories, including regular semigroups, monoids, and groups, as well as less conventional semigroup classes, such as inverse semigroups, transformative semigroups, and idempotent semigroups. By delving into the intricacies of each semigroup type's ideal structures, this work seeks to expand the current understanding of these algebraic structures, offering new insights and potential applications in various fields, such as computer science, mathematics, and theoretical physics.

BackgroundA genetic network can be represented as a directed graph in which a node corresponds to a gene and a directed edge specifies the direction of influence of one gene on another. The reconstruction of such networks from transcript profiling data remains an important yet challenging endeavor. A transcript profile specifies the abundances of many genes in a biological sample of interest. Prevailing strategies for learning the structure of a genetic network from high-dimensional transcript profiling data assume sparsity and linearity. Many methods consider relatively small directed graphs, inferring graphs with up to a few hundred nodes. This work examines large undirected graphs representations of genetic networks, graphs with many thousands of nodes where an undirected edge between two nodes does not indicate the direction of influence, and the problem of estimating the structure of such a sparse linear genetic network (SLGN) from transcript profiling data.ResultsThe structure learning task is cast as a sparse linear regression problem which is then posed as a LASSO (l1-constrained fitting) problem and solved finally by formulating a Linear Program (LP). A bound on the Generalization Error of this approach is given in terms of the Leave-One-Out Error. The accuracy and utility of LP-SLGNs is assessed quantitatively and qualitatively using simulated and real data. The Dialogue for Reverse Engineering Assessments and Methods (DREAM) initiative provides gold standard data sets and evaluation metrics that enable and facilitate the comparison of algorithms for deducing the structure of networks. The structures of LP-SLGNs estimated from the INSILICO1, INSILICO2 and INSILICO3 simulated DREAM2 data sets are comparable to those proposed by the first and/or second ranked teams in the DREAM2 competition. The structures of LP-SLGNs estimated from two published Saccharomyces cerevisae cell cycle transcript profiling data sets capture known regulatory associations. In each S. cerevisiae LP-SLGN, the number of nodes with a particular degree follows an approximate power law suggesting that its degree distributions is similar to that observed in real-world networks. Inspection of these LP-SLGNs suggests biological hypotheses amenable to experimental verification.ConclusionA statistically robust and computationally efficient LP-based method for estimating the topology of a large sparse undirected graph from high-dimensional data yields representations of genetic networks that are biologically plausible and useful abstractions of the structures of real genetic networks. Analysis of the statistical and topological properties of learned LP-SLGNs may have practical value; for example, genes with high random walk betweenness, a measure of the centrality of a node in a graph, are good candidates for intervention studies and hence integrated computational – experimental investigations designed to infer more realistic and sophisticated probabilistic directed graphical model representations of genetic networks. The LP-based solutions of the sparse linear regression problem described here may provide a method for learning the structure of transcription factor networks from transcript profiling and transcription factor binding motif data.

In this article, a general approach for directed graph clustering and two new density-based clustering objectives are presented. First, using an equivalence between the clustering objective functions and a trace maximization expression, the directed graph clustering objectives are converted into the corresponding weighted kernel <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -means problems. Then, a nonspectral algorithm, which covers both the direction and weight information of the directed graphs, is thus proposed. Next, with Rayleigh’s quotient, the upper and lower bounds of clustering objectives are obtained. After that, we introduce a new definition of weak links to characterize the effectiveness of clustering. Finally, illustrative examples are given to demonstrate effectiveness of the results. This article provides a glance at the potential connection between density-based and pattern-based clustering. Compared with other approaches for directed graph clustering, the method proposed in this article naturally avoids the loss of the nonsymmetric edge data because there is no need for any additional symmetrization.