@article{Donganont2025, 
author = {Siriluk Donganont and Sun Shin Ahn and Hee Sik Kim},
title = {Several co-associative laws and pre-   B-algebras},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {4},
pages = {9332-9341},
keywords = {pre-B-algebra, B-algebra, d-algebra, co-associative, ⊥-algebra, anti-commutative, post groupoid, pre-semigroup},
url = {https://www.sciopen.com/article/10.3934/math.2025431},
doi = {10.3934/math.2025431},
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    ⊥-algebra. We apply the notions of a post groupoid and a pre-semigroup of a groupoid to the set    (      N    ,  +  ) of all nonnegative integers, and we prove that the groupoid    (      N    ,  +  ) cannot be a post groupoid of a    B-algebra or an edge    d-algebra.}
}