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Keywords = λ-semidirect product

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16 pages, 278 KiB  
Article
Zappa–Szép Groupoids of Inverse Semigroups and an Alternative Proof of Billhardt’s λ-Semidirect Products
by Suha Wazzan
Mathematics 2025, 13(7), 1122; https://doi.org/10.3390/math13071122 - 28 Mar 2025
Viewed by 199
Abstract
The aim of this paper is to introduce and study Zappa-Szép groupoids of inverse semigroups. Some properties of such kinds of groupoids are explored. As an application, an alternative proof of Billhardt’s λ-semidirect products is given. We finish with several examples that [...] Read more.
The aim of this paper is to introduce and study Zappa-Szép groupoids of inverse semigroups. Some properties of such kinds of groupoids are explored. As an application, an alternative proof of Billhardt’s λ-semidirect products is given. We finish with several examples that highlight the versatility and applicability of Zappa-Szép groupoids in various types of inverse semigroups. Full article
(This article belongs to the Special Issue Theory and Application of Algebraic Combinatorics)
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22 pages, 341 KiB  
Article
Jacobi–Jordan Conformal Algebras: Basics, Constructions and Related Structures
by Taoufik Chtioui, Sami Mabrouk and Abdenacer Makhlouf
Mathematics 2025, 13(5), 843; https://doi.org/10.3390/math13050843 - 3 Mar 2025
Viewed by 710
Abstract
The main purpose of this paper is to introduce and investigate the notion of Jacobi–Jordan conformal algebras. They are a generalization of Jacobi–Jordan algebras which correspond to the case in which the formal parameter λ equals 0. We consider some related structures such [...] Read more.
The main purpose of this paper is to introduce and investigate the notion of Jacobi–Jordan conformal algebras. They are a generalization of Jacobi–Jordan algebras which correspond to the case in which the formal parameter λ equals 0. We consider some related structures such as conformal modules, corresponding representations and O-operators. Therefore, conformal derivations from Jacobi–Jordan conformal algebras to their conformal modules are used to describe conformal derivations of Jacobi–Jordan conformal algebras of the semidirect product type. Moreover, we study a class of Jacobi–Jordan conformal algebras called quadratic Jacobi–Jordan conformal algebras, which are characterized by mock-Gel’fand–Dorfman bialgebras. Finally, the C[]-split extending structures problem for Jacobi–Jordan conformal algebras is studied. Furthermore, we introduce an unified product of a given Jacobi–Jordan conformal algebra J and a given C[]-module K. This product includes some other interesting products of Jacobi–Jordan conformal algebras such as the twisted product and crossed product. Using this product, a cohomological type object is constructed to provide a theoretical answer to the C[]-split extending structures problem. Full article
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