Graph Inverse Semigroups Research Articles

Properties of congruence lattices of graph inverse semigroups

From any directed graph [Formula: see text] one can construct the graph inverse semigroup [Formula: see text], whose elements, roughly speaking, correspond to paths in [Formula: see text]. Wang and Luo showed that the congruence lattice [Formula: see text] of [Formula: see text] is upper-semimodular for every graph [Formula: see text], but can fail to be lower-semimodular for some [Formula: see text]. We provide a simple characterization of the graphs [Formula: see text] for which [Formula: see text] is lower-semimodular. We also describe those [Formula: see text] such that [Formula: see text] is atomistic, and characterize the minimal generating sets for [Formula: see text] when [Formula: see text] is finite and simple.

Leavitt inverse semigroups of polynomial growth

We relate growth functions of graph inverse semigroups, Leavitt inverse semigroups and Leavitt path algebras and discuss structure of Leavitt inverse semigroups of polynomially bounded growth.

Quasi-Semilattices on Networks

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.

Distributivity in congruence lattices of graph inverse semigroups

Let be a directed graph and be the graph inverse semigroup of . Luo and Wang (Semimodularity in congruence lattice of graph inverse semigroups, 2021) showed that the congruence lattice of any graph inverse semigroup is upper semimodular, but not lower semimodular in general. Anagnostopoulou-Merkouri, Mesyan and Mitchell (Properties of congruence lattices of graph inverse semigroups, 2022) characterized the directed graph for which is lower semimodular. In the present paper, we show that lower semimodularity, modularity and distributivity in the congruence lattice of any graph inverse semigroup are equivalent.

On a class of inverse semigroups related to Leavitt path algebras

We introduce a class of inverse semigroups built from directed graphs that we refer to as Leavitt inverse semigroups. These semigroups are closely related to graph inverse semigroups and Leavitt path algebras. We find a presentation for the Leavitt inverse semigroup of a graph in terms of generators and relations. We describe the structure of the Leavitt inverse semigroup and the Leavitt path algebra of a graph that admits a directed immersion into a circle. We show that two graphs that have isomorphic Leavitt inverse semigroups have isomorphic Leavitt path algebras and we classify graphs that have isomorphic Leavitt inverse semigroups. As a consequence, we show that Leavitt path algebras are 0-retracts of certain matrix algebras.

Semimodularity in congruence lattices of graph inverse semigroups

Congruences on a graph inverse semigroup were recently described in terms of the underline graph. Based on such descriptions, we show that the congruence lattice of a graph inverse semigroup is upper semimodular but not lower semimodular.

On graph inverse semigroups

We study the relationship between the graph inverse semigroups of two graphs when there is a directed immersion between the graphs and we provide structural information about graph inverse semigroups of finite graphs that admit a directed cover onto a bouquet of circles. We provide a topological characterization of the universal groups of the local submonoids of a graph inverse semigroup. We also find necessary and sufficient conditions for a homomorphic image of a graph inverse semigroup to be another graph inverse semigroup.

Embedding of graph inverse semigroups into CLP-compact topological semigroups

In this paper we investigate graph inverse semigroups which are subsemigroups of compact-like topological semigroups. More precisely, we characterize graph inverse semigroups which admit a compact semigroup topology and describe graph inverse semigroups which can be embedded densely into CLP-compact topological semigroups.

On locally compact topological graph inverse semigroups

In this paper we characterize graph inverse semigroups which admit only discrete locally compact semigroup topology. This characterization provides a complete answer to a question of Z. Mesyan, J. D. Mitchell, M. Morayne and Y. H. Péresse posed in [26].

Congruences on graph inverse semigroups

Graph inverse semigroups were defined by Ash and Hall in 1975. They found necessary and sufficient conditions for the semigroups to be congruence free. In this paper we give a classification of congruences on a graph inverse semigroup in terms of the underlying graph. As a consequence, we show that the inverse graph semigroup of a finite graph is congruence Noetherian.

An alternative look at the structure of graph inverse semigroups

For any graph inverse semigroup $G(E)$ we describe subsemigroups $D^0=D\cup\{0\}$ and $J^0=J\cup\{0\}$ of $G(E)$ where $D$ and $J$ are arbitrary $\mathcal{D}$-class and $\mathcal{J}$-class of $G(E)$, respectively. In particular, we prove that for each $\mathcal{D}$-class $D$ of a graph inverse semigroup over an acyclic graph the semigroup $D^0$ is isomorphic to a semigroup of matrix units. Also we show that for any elements $a,b$ of a graph inverse semigroup $G(E)$, $J_a\cdot J_b\cup J_b\cdot J_a\subset J_b^0$ if there exists a path $w$ such that $s(w)\in J_a$ and $r(w)\in J_b$.

On universal objects in the class of graph inverse semigroups

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 $${\mathscr {P}}_{\lambda }$$ where $$\lambda =|G(E)|$$. We show that each graph inverse semigroup G(E) admits the coarsest inverse semigroup topology $$\tau $$. Moreover, each injective homomorphism from $$(G(E),\tau )$$ to $$({\mathscr {P}}_{|G(E)|},\tau )$$ is a topological embedding.

On locally compact semitopological graph inverse semigroups

In this paper we investigate locally compact semitopological graph inverse semigroups. Our main result is the following: if a directed graph $E$ is strongly connected and contains a finite amount of vertices then a locally compact semitopological graph inverse semigroup $G(E)$ is either compact or discrete. This result generalizes results of Gutik and Bardyla who proved the above dichotomy for locally compact semitopological polycyclic monoids $\mathcal{P}_1$ and $\mathcal{P}_{\lambda}$, respectively.

Closed inverse subsemigroups of graph inverse semigroups

ABSTRACTAs part of his study of representations of the polycylic monoids, Lawson described all the closed inverse submonoids of a polycyclic monoid Pn and classified them up to conjugacy. We show that Lawson’s description can be extended to closed inverse subsemigroups of graph inverse semigroups. We then apply Schein’s theory of cosets in inverse semigroups to the closed inverse subsemigroups of graph inverse semigroups: we give necessary and sufficient conditions for a closed inverse subsemigroup of a graph inverse semigroup to have finite index, and determine the value of the index when it is finite.

Topological graph inverse semigroups

To every directed graph E one can associate a graph inverse semigroupG(E), where elements roughly correspond to possible paths in E. These semigroups generalize polycyclic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, graph C⁎-algebras, and Toeplitz C⁎-algebras. We investigate topologies that turn G(E) into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, G(E)∖{0} must be discrete for any directed graph E. On the other hand, G(E) need not be discrete in a Hausdorff semigroup topology, and for certain graphs E, G(E) admits a T1 semigroup topology in which G(E)∖{0} is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of G(E) in larger topological semigroups.

The structure of a graph inverse semigroup

Given any directed graph E one can construct a graph inverse semigroup G(E), where, roughly speaking, elements correspond to paths in the graph. In this paper we study the semigroup-theoretic structure of G(E). Specifically, we describe the non-Rees congruences on G(E), show that the quotient of G(E) by any Rees congruence is another graph inverse semigroup, and classify the G(E) that have only Rees congruences. We also find the minimum possible degree of a faithful representation by partial transformations of any countable G(E), and we show that a homomorphism of directed graphs can be extended to a homomorphism (that preserves zero) of the corresponding graph inverse semigroups if and only if it is injective.

On subshift presentations

We consider partitioned graphs, by which we mean finite directed graphs with a partitioned edge set${\mathcal{E}}={\mathcal{E}}^{-}\cup {\mathcal{E}}^{+}$. Additionally given a relation${\mathcal{R}}$between the edges in${\mathcal{E}}^{-}$and the edges in${\mathcal{E}}^{+}$, and under the appropriate assumptions on${\mathcal{E}}^{-},{\mathcal{E}}^{+}$and${\mathcal{R}}$, denoting the vertex set of the graph by$\mathfrak{P}$, we speak of an${\mathcal{R}}$-graph${\mathcal{G}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$. From${\mathcal{R}}$-graphs${\mathcal{G}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$we construct semigroups (with zero)${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$that we call${\mathcal{R}}$-graph semigroups. We write a list of conditions on a topologically transitive subshift with property$(A)$that together are sufficient for the subshift to have an${\mathcal{R}}$-graph semigroup as its associated semigroup.Generalizing previous constructions, we describe a method of presenting subshifts by means of suitably structured finite labeled directed graphs$({\mathcal{V}},~\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706}~)$with vertex set${\mathcal{V}}$, edge set$\unicode[STIX]{x1D6F4}$, and a label map that assigns to the edges in$\unicode[STIX]{x1D6F4}$labels in an${\mathcal{R}}$-graph semigroup${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{-})$. We denote the presented subshift by$X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$and call$X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$an${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{-})$-presentation.We introduce a property$(B)$of subshifts that describes a relationship between contexts of admissible words of a subshift, and we introduce a property$(c)$of subshifts that in addition describes a relationship between the past and future contexts and the context of admissible words of a subshift. Property$(B)$and the simultaneous occurrence of properties$(B)$and$(c)$are invariants of topological conjugacy.We consider subshifts in which every admissible word has a future context that is compatible with its entire past context. Such subshifts we call right instantaneous. We introduce a property$RI$of subshifts, and we prove that this property is a necessary and sufficient condition for a subshift to have a right instantaneous presentation. We consider also subshifts in which every admissible word has a future context that is compatible with its entire past context, and also a past context that is compatible with its entire future context. Such subshifts we call bi-instantaneous. We introduce a property$BI$of subshifts, and we prove that this property is a necessary and sufficient condition for a subshift to have a bi-instantaneous presentation.We define a subshift as strongly bi-instantaneous if it has for every sufficiently long admissible word$a$an admissible word$c$, that is contained in both the future context of$a$and the past context of$a$, and that is such that the word$ca$is a word in the future context of$a$that is compatible with the entire past context of$a$, and the word$ac$is a word in the past context of$a$, that is compatible with the entire future context of$a$. We show that a topologically transitive subshift with property$(A)$, and associated semigroup a graph inverse semigroup${\mathcal{S}}$, has an${\mathcal{S}}$-presentation, if and only if it has properties$(c)$and$BI$, and a strongly bi-instantaneous presentation, if and only if it has properties$(c)$and$BI$, and all of its bi-instantaneous presentations are strongly bi-instantaneous.We construct a class of subshifts with property$(A)$, to which certain graph inverse semigroups${\mathcal{S}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$are associated, that do not have${\mathcal{S}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$-presentations.We associate to the labeled directed graphs$({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$topological Markov chains and Markov codes, and we derive an expression for the zeta function of$X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$in terms of the zeta functions of the topological Markov shifts and the generating functions of the Markov codes.

The $K$-Theory of Some Reduced Inverse Semigroup $C^*$-Algebras

We use a recent result by Cuntz, Echterhoff and Li about the $K$-theory of certain reduced $C^*$-crossed products to describe the $K$-theory of $C^*_r(S)$ when $S$ is an inverse semigroup satisfying certain requirements. A result of Milan and Steinberg allows us to show that $C^*_r(S)$ is Morita equivalent to a crossed product of the type handled by Cuntz, Echterhoff and Li. We apply our result to graph inverse semigroups and the inverse semigroups of one-dimensional tilings.

TRACES ON SEMIGROUP RINGS AND LEAVITT PATH ALGEBRAS

AbstractThe trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.

Graph inverse semigroups: Their characterization and completion

Graph inverse semigroups generalize the polycyclic inverse monoids and play an important role in the theory of C⁎-algebras. In this paper, we characterize such semigroups and show how they may be completed, under suitable conditions, to form what we call the Cuntz–Krieger semigroup of the graph. This semigroup is proved to be the ample semigroup of a topological groupoid associated with the graph, and the semigroup analogue of the Leavitt path algebra of the graph.