Several co-associative laws and pre-$ B $-algebras

Siriluk Donganont; Sun Shin Ahn; Hee Sik Kim; Siriluk Donganont; Sun Shin Ahn; Hee Sik Kim

doi:10.3934/math.2025431

AIMS Mathematics

2025, Volume 10, Issue 4: 9332-9341. doi: 10.3934/math.2025431

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Several co-associative laws and pre-$ B $-algebras

1.
School of Sciences, University of Phayao, Phayao 56000, Thailand
2.
Department of Mathematics Education, Dongguk University, Seoul 04620, Korea
3.
Department of Mathematics, Hanyang University, Seoul 04763, Korea

Received: 11 September 2024 Revised: 31 March 2025 Accepted: 17 April 2025 Published: 23 April 2025
MSC : 06F35, 20N02

In this paper, we introduce several co-associative laws and the notion of a pre-$ B $-algebra. We show that every $ B $-algebra is both a pre-$ B $-algebra and a $ \perp $-algebra. We apply the notions of a post groupoid and a pre-semigroup of a groupoid to the set $ (\mathbb{N}, +) $ of all nonnegative integers, and we prove that the groupoid $ (\mathbb{N}, +) $ cannot be a post groupoid of a $ B $-algebra or an edge $ d $-algebra.
- pre-$ B $-algebra,
- $ B $-algebra,
- $ d $-algebra,
- co-associative,
- $ \perp $-algebra,
- anti-commutative,
- post groupoid,
- pre-semigroup
Citation: Siriluk Donganont, Sun Shin Ahn, Hee Sik Kim. Several co-associative laws and pre-$ B $-algebras[J]. AIMS Mathematics, 2025, 10(4): 9332-9341. doi: 10.3934/math.2025431

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Abstract

In this paper, we introduce several co-associative laws and the notion of a pre-$ B $-algebra. We show that every $ B $-algebra is both a pre-$ B $-algebra and a $ \perp $-algebra. We apply the notions of a post groupoid and a pre-semigroup of a groupoid to the set $ (\mathbb{N}, +) $ of all nonnegative integers, and we prove that the groupoid $ (\mathbb{N}, +) $ cannot be a post groupoid of a $ B $-algebra or an edge $ d $-algebra.

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