scholarly journals Graph varieties containing Murskii's groupoid

1989 ◽  
Vol 39 (2) ◽  
pp. 265-276
Author(s):  
R. Pöschel
Keyword(s):  
Finitely Based ◽  
Graph Algebras ◽  
Undirected Graphs ◽  
Graph Algebra ◽  
Subvariety Lattice ◽  
As Graph ◽  
Graph Varieties ◽  
Nonfinitely Based

In this paper varieties are investigated which are generated by graph algebras of undirected graphs and—in most cases—contain Murskii's groupoid (that is the graph algebra of the graph with two adjacent vertices and one loop). Though these varieties are inherently nonfinitely based, they can be finitely based as graph varieties (finitely graph based) like, for example, the varitey generated by Murskii's groupoid. Many examples of nonfinitely based graph varities containing Murskii's groupoid are given, too. Moreover, the coatoms in the subvariety lattice of the graph variety of all undirected graphs are described. There are two coatoms and they are finitely graph based.

IDENTITIES IN BIREGULAR LEFTMOST GRAPH VARIETIES OF TYPE (2,0)

2009 ◽  
Vol 02 (01) ◽  
pp. 1-17
Author(s):  
Apinant Anantpiniwatna ◽  
Tiang Poomsa-Ard
Keyword(s):  
Universal Algebra ◽  
Directed Graphs ◽  
Graph Algebras ◽  
Universal Algebras ◽  
Graph Algebra ◽  
Basic Concepts ◽  
Multiple Edges ◽  
Graph Varieties

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies a term equation s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A class of graph algebras V is called a graph variety if V = ModgΣ where Σ is a subset of T(X) × T(X). A graph variety V' = ModgΣ' is called a biregular leftmost graph variety if Σ' is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity in a variety V if G satisfies s ≈ t for all G ∈ V. In this paper we characterize identities in each biregular leftmost graph variety. For identities, varieties and other basic concepts of universal algebra see e.g. [1].

Graph variety generated by linear terms

2019 ◽  
Vol 12 (05) ◽  
pp. 1950074
Author(s):  
C. Manyuen ◽  
P. Jampachon ◽  
T. Poomsa-ard
Keyword(s):  
Directed Graphs ◽  
Linear Term ◽  
Graph Algebras ◽  
Universal Algebras ◽  
Type Formula ◽  
Graph Algebra ◽  
Multiple Edges ◽  
Graph Varieties

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type [Formula: see text]. We say that a graph [Formula: see text] satisfies a term equation [Formula: see text] if the corresponding graph algebra [Formula: see text] satisfies [Formula: see text]. The set of all term equations [Formula: see text], which the graph [Formula: see text] satisfies, is denoted by [Formula: see text]. The class of all graph algebras satisfy all term equations in [Formula: see text] is called the graph variety generated by [Formula: see text] denoted by [Formula: see text]. A term is called a linear term if each variable which occurs in the term, occurs only once. A term equation [Formula: see text] is called a linear term equation if [Formula: see text] and [Formula: see text] are linear terms. This paper is devoted to a thorough investigation of graph varieties defined by linear term equations. In particular, we give a complete description of rooted graphs generating a graph variety described by linear term equations.

scholarly journals Associative spectra of graph algebras I

2021 ◽  
Author(s):  
Erkko Lehtonen ◽  
Tamás Waldhauser
Keyword(s):  
Catalan Numbers ◽  
Graph Algebras ◽  
Undirected Graphs ◽  
Powers Of 2

AbstractAssociative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities: constant 1, powers of 2, and Catalan numbers. Associative and antiassociative digraphs are described, and associative spectra are determined for certain families of digraphs, such as paths, cycles, and graphs on two vertices.

scholarly journals Graph algebras and orbit equivalence

2015 ◽  
Vol 37 (2) ◽  
pp. 389-417 ◽  
Author(s):  
NATHAN BROWNLOWE ◽  
TOKE MEIER CARLSEN ◽  
MICHAEL F. WHITTAKER
Keyword(s):  
Directed Graphs ◽  
Orbit Equivalence ◽  
Graph Algebras ◽  
Orbit Equivalent ◽  
Markov Shifts ◽  
Graph Algebra ◽  
Arbitrary Graphs ◽  
Reverse Implication

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto’s notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their $C^{\ast }$-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs $E$ we construct a groupoid ${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$ from the graph algebra $C^{\ast }(E)$ and its diagonal subalgebra ${\mathcal{D}}(E)$ which generalises Renault’s Weyl groupoid construction applied to $(C^{\ast }(E),{\mathcal{D}}(E))$. We show that ${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$ recovers the graph groupoid ${\mathcal{G}}_{E}$ without the assumption that every cycle in $E$ has an exit, which is required to apply Renault’s results to $(C^{\ast }(E),{\mathcal{D}}(E))$. We finish with applications of our results to out-splittings of graphs and to amplified graphs.

The finite basis property of a certain semigroup of upper triangular matrices over a field

2016 ◽  
Vol 15 (09) ◽  
pp. 1650177 ◽  
Author(s):  
Yuzhu Chen ◽  
Xun Hu ◽  
Yanfeng Luo
Keyword(s):  
Basis Property ◽  
Finite Basis ◽  
Main Diagonal ◽  
Finitely Based ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Finite Basis Property ◽  
Open Question ◽  
Nonfinitely Based ◽  
Secondary Diagonal

Let [Formula: see text] be the semigroup of all upper triangular [Formula: see text] matrices over a field [Formula: see text] whose main diagonal entries are [Formula: see text]s and/or [Formula: see text]s. Volkov proved that [Formula: see text] is nonfinitely based as both a plain semigroup and an involution semigroup under the reflection with respect to the secondary diagonal. In this paper, we shall prove that [Formula: see text] is finitely based for any field [Formula: see text]. When [Formula: see text], this result partially answers an open question posed by Volkov.

scholarly journals -algebras of labelled graphs III—-theory computations

2015 ◽  
Vol 37 (2) ◽  
pp. 337-368 ◽  
Author(s):  
TERESA BATES ◽  
TOKE MEIER CARLSEN ◽  
DAVID PASK
Keyword(s):  
Uniqueness Theorem ◽  
Inductive Limit ◽  
Important Class ◽  
Graph Algebras ◽  
Gauge Invariant ◽  
Shift Spaces ◽  
Graph Algebra ◽  
Common Framework ◽  
Definition Of ◽  
Labelled Graphs

In this paper we give a formula for the$K$-theory of the$C^{\ast }$-algebra of a weakly left-resolving labelled space. This is done by realizing the$C^{\ast }$-algebra of a weakly left-resolving labelled space as the Cuntz–Pimsner algebra of a$C^{\ast }$-correspondence. As a corollary, we obtain a gauge-invariant uniqueness theorem for the$C^{\ast }$-algebra of any weakly left-resolving labelled space. In order to achieve this, we must modify the definition of the$C^{\ast }$-algebra of a weakly left-resolving labelled space. We also establish strong connections between the various classes of$C^{\ast }$-algebras that are associated with shift spaces and labelled graph algebras. Hence, by computing the$K$-theory of a labelled graph algebra, we are providing a common framework for computing the$K$-theory of graph algebras, ultragraph algebras, Exel–Laca algebras, Matsumoto algebras and the$C^{\ast }$-algebras of Carlsen. We provide an inductive limit approach for computing the$K$-groups of an important class of labelled graph algebras, and give examples.

scholarly journals Realizations of AF-algebras as graph algebras, Exel–Laca algebras, and ultragraph algebras

2009 ◽  
Vol 257 (5) ◽  
pp. 1589-1620 ◽  
Author(s):  
Takeshi Katsura ◽  
Aidan Sims ◽  
Mark Tomforde
Keyword(s):  
Graph Algebras ◽  
As Graph ◽  
Af Algebras

Graph algebras and graph varieties

1990 ◽  
Vol 27 (4) ◽  
pp. 559-577 ◽  
Author(s):  
Reinhard P�schel
Keyword(s):  
Graph Algebras ◽  
Graph Varieties

scholarly journals Deformations of Fell bundles and twisted graph algebras

2016 ◽  
Vol 161 (3) ◽  
pp. 535-558 ◽  
Author(s):  
IAIN RAEBURN
Keyword(s):  
Discrete Groups ◽  
Graph Algebras ◽  
Graph Algebra

AbstractWe consider Fell bundles over discrete groups, and theC*-algebra which is universal for representations of the bundle. We define deformations of Fell bundles, which are new Fell bundles with the same underlying Banach bundle but with the multiplication deformed by a two-cocycle on the group. Every graph algebra can be viewed as theC*-algebra of a Fell bundle, and there are many cocycles of interest with which to deform them. We thus obtain many of the twisted graph algebras of Kumjian, Pask and Sims. We demonstate the utility of our approach to these twisted graph algebras by proving that the deformations associated to different cocycles can be assembled as the fibres of aC*-bundle.

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