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16 pages, 299 KiB

Open AccessArticle

On the Equational Theory of Lattice-Based Algebras for Layered Graphs

by Zhe Yu, Hao Zhan, Yiheng Wang, Zhe Lin and Fei Liang

Axioms 2025, 14(4), 257; https://doi.org/10.3390/axioms14040257 - 28 Mar 2025

Viewed by 290

Layered algebras are introduced and used to express layered graphs. Layered graphs are considered to be a highly effective abstract tool to manage the difficulty in conceptualizing and reasoning regarding complex systems related to coding in email exchange and access control in security. In the present paper, we study the varieties of several classes of lattice-based layer algebras and show that all these varieties have decidable equational theory via a finite model property. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications, 2nd Edition)

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34 pages, 403 KiB

Open AccessArticle

Internal Categorical Structures and Their Applications

by Nelson Martins-Ferreira

Mathematics 2023, 11(3), 660; https://doi.org/10.3390/math11030660 - 28 Jan 2023

Cited by 1 | Viewed by 1639

Abstract

While surveying some internal categorical structures and their applications, it is shown that triangulations and internal groupoids can be unified as two different instances of the same common structure, namely a multi-link. A brief survey includes the categories of directed graphs, reflexive graphs, links, multi-links, triangulations, trigraphs, multiplicative graphs, groupoids, pregroupoids, internal categories, kites, directed kites and multiplicative kites. Most concepts are well-known, and all of them have appeared in print at least once. For example, a multiplicative directed kite has been used as a common generalization for an internal category and a pregroupoid. The scope of the notion of centralization for equivalence relations is widened into the context of digraphs while providing a new characterization of internal groupoids. Full article

(This article belongs to the Special Issue Mathematical Structures and Their Applications)

55 pages, 544 KiB

Open AccessArticle

An Index for Graphs and Graph Groupoids

by Ilwoo Cho and Palle Jorgensen

Axioms 2022, 11(2), 47; https://doi.org/10.3390/axioms11020047 - 25 Jan 2022

Cited by 3 | Viewed by 2720

Abstract

In this paper, we consider certain quantities that arise in the images of the so-called graph-tree indexes of graph groupoids. In text, the graph groupoids are induced by connected finite-directed graphs with more than one vertex. If a graph groupoid

G

G contains [...] Read more.

G

G contains at least one loop-reduced finite path, then the order of

G

is infinity; hence, the canonical groupoid index

[G : K]

of the inclusion

K \subseteq G

is either ∞ or 1 (under the definition and a natural axiomatization) for the graph groupoids

K

of all “parts” K of G. A loop-reduced finite path generates a semicircular element in graph groupoid algebra. Thus, the existence of semicircular systems acting on the free-probabilistic structure of a given graph G is guaranteed by the existence of loop-reduced finite paths in

G

. The non-semicircularity induced by graphs yields a new index-like notion called the graph-tree index

Γ

G

. We study the connections between our graph-tree index and non-semicircular cases. Hence, non-semicircularity also yields the classification of our graphs in terms of a certain type of trees. As an application, we construct towers of graph-groupoid-inclusions which preserve the graph-tree index. We further show that such classification applies to monoidal operads. Full article

(This article belongs to the Collection Mathematical Analysis and Applications)

10 pages, 281 KiB

Open AccessArticle

Global Symmetries, Local Symmetries and Groupoids

by Michel Petitjean

Symmetry 2021, 13(10), 1905; https://doi.org/10.3390/sym13101905 - 10 Oct 2021

Cited by 2 | Viewed by 4459

Abstract

Local symmetries are primarily defined in the case of spacetime, but several authors have defined them outside this context, sometimes with the help of groupoids. We show that, in many cases, local symmetries can be defined as global symmetries. We also show that groups can be used, rather than groupoids, to handle local symmetries. Examples are given for graphs and networks, color symmetry and tilings. The definition of local symmetry in physics is also discussed. Full article

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11 pages, 301 KiB

Open AccessArticle

A Method to Identify Simple Graphs by Special Binary Systems

by Hee Sik Kim, J. Neggers and Sun Shin Ahn

Symmetry 2018, 10(7), 297; https://doi.org/10.3390/sym10070297 - 23 Jul 2018

Cited by 4 | Viewed by 2976

Abstract

In this paper, we discuss some relations between the semigroup,

B i n (X)

, of all groupoids

(X, *)

and graphs. We discuss mimimum (mutual) covering sets in several groupoids and discuss distances of graphs with groupoids. [...] Read more.

In this paper, we discuss some relations between the semigroup,

B i n (X)

, of all groupoids

(X, *)

and graphs. We discuss mimimum (mutual) covering sets in several groupoids and discuss distances of graphs with groupoids. Finally, we obtain some results on frame graphs with groupoids. Full article

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