A strong geometric hyperbolicity property for directed graphs and monoids

Multiplication tables and word-hyperbolicity in free products of semigroups, monoids, and groups

This article studies the properties of word-hyperbolic semigroups and monoids, i.e. those having context-free multiplication tables with respect to a regular combing, as defined by Duncan & Gilman. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied e.g. by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied…

Finite presentability and isomorphism of Cayley graphs of monoids

Two finitely generated monoids are constructed, one finitely presented the other not, whose (directed, unlabelled) Cayley graphs are isomorphic.

Classification of Label-Regular Directed Trees up to Almost Isomorphism

This paper outlines a method to determine whether two label-regular directed trees, are isomorphic and when they are almost isomorphic. The approach involves reinterpreting label-regular directed trees as universal covers of rooted graphs. This allows us associate a unique graph with each isomorphism class of a label-regular directed tree. Additionally, by examining the graph monoid we can verify when two unfolding graphs produce almost isomorphic unfolding trees, thereby classifying unfolding trees up to almost isomorphism.

A graph-dynamical interpretation of Kiselman's semigroups

A Sequential Dynamical System (SDS) is a quadruple (Γ, S_i,f_i,w) consisting of a (directed) graph Γ=(V,E), each of whose vertices i\in V is endowed with a finite set state S_i and an update function f_i: \prod_{j, i \to j} S_j \to S_i --- we call this structure an {\em update system} --- and a word w in the free monoid over V, specifying the order in which update functions are to be performed. Each word induces an evolution of the system and in this paper we are interested in the dynamics monoid, whose elements are all possible evolutions. When Γis a directed acyclic graph, the dynamics monoi…

On universal objects in the class of graph inverse semigroups

In this paper we show that polycyclic monoids are universal objects in the class of graph inverse semigroups. In particular, we prove that a graph inverse semigroup $G(E)$ over a directed graph $E$ embeds into the polycyclic monoid $\mathcal{P}_λ$ where $λ=|G(E)|$. We show that each graph inverse semigroup $G(E)$ admits the coarsest inverse semigroup topology $τ$. Moreover, each injective homomorphism from $(G(E),τ)$ to the $(\mathcal{P}_{|G(E)|},τ)$ is a topological embedding.

有限群的自同态图与自同构图

Endomorphism and automorphism graphs of finite groups

本文定义有限群的有向/无向自同态图与自同构图，介绍其与变换幺半群的关联，探究其图论性质、判定对应图类的群类型并计算特殊群实例，最后以示例说明相关问题的复杂性。