$ N(2, 2, 0) $-Algebras and dimonoids

Fang-An Deng; Qiao Xin; Qian Yang; Suixin He; Yichuan Yang; Fang-An Deng; Qiao Xin; Qian Yang; Suixin He; Yichuan Yang

doi:10.3934/math.20261003

AIMS Mathematics

2026, Volume 11, Issue 8: 24958-24977. doi: 10.3934/math.20261003

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Research article

$ N(2, 2, 0) $-Algebras and dimonoids

1.
School of Mathematics and Statistics, Yili Normal University, Yili 835000, China
2.
School of Mathematical Sciences, Beihang University, Shahe Campus, Beijing 102206, China

Received: 27 May 2026 Revised: 12 July 2026 Accepted: 21 July 2026 Published: 13 August 2026
MSC : 03G25, 06F35, 08A05, 18B40, 20M10

J.-L. Loday introduced the concept of dimonoids, which serve as fundamental tools for investigating Leibniz algebras. $ N(2, 2, 0) $-algebras, abbreviated as NA-algebras, are bisemigroup algebraic systems equipped with two associative binary operations subject to extra identities. This work constructs a novel research paradigm to advance theoretical and applied studies of NA-algebras from the perspective of dimonoid theory. By thoroughly exploring the left and right commutative semigroup properties of NA-algebras, we reveal the inherent relationships between NA-algebras and a range of related algebraic structures: dimonoids, doppelsemigroups, $ g $-dimonoids, restrictive bisemigroups, and digroups. We derive sufficient conditions under which these structures may carry an NA-algebra structure, and prove that every NA-algebra is an abelian dimonoid with a bar-unit denoted "1".
- $ N(2, 2, 0) $-algebras,
- dimonoid,
- semigroup,
- digroups,
- doppelsemigroups
Citation: Fang-An Deng, Qiao Xin, Qian Yang, Suixin He, Yichuan Yang. $ N(2, 2, 0) $-Algebras and dimonoids[J]. AIMS Mathematics, 2026, 11(8): 24958-24977. doi: 10.3934/math.20261003

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Abstract

J.-L. Loday introduced the concept of dimonoids, which serve as fundamental tools for investigating Leibniz algebras. $ N(2, 2, 0) $-algebras, abbreviated as NA-algebras, are bisemigroup algebraic systems equipped with two associative binary operations subject to extra identities. This work constructs a novel research paradigm to advance theoretical and applied studies of NA-algebras from the perspective of dimonoid theory. By thoroughly exploring the left and right commutative semigroup properties of NA-algebras, we reveal the inherent relationships between NA-algebras and a range of related algebraic structures: dimonoids, doppelsemigroups, $ g $-dimonoids, restrictive bisemigroups, and digroups. We derive sufficient conditions under which these structures may carry an NA-algebra structure, and prove that every NA-algebra is an abelian dimonoid with a bar-unit denoted "1".

References

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