On finite \(B\)-algebra.
type: Item
instance of: scholarly article
title: On finite \(B\)-algebra.
zbMATH Open document ID: 1334.06034
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Afrika Matematika
publication date: 2015-09-23T00:00:00Z
review text: The authors consider properties of \(B\)-algebras, especially finite \(B\)-algebras and show some results about them. For example, they prove that there is no proper \(B\)-algebra \(X\) such that \(|X|<6\) (Theorem 4.2). An algebraic structure \((X,*,0)\) of type \((2,0)\) is called a \(B\)-algebra if it satisfies the following axioms: 1. \(x*0=x\); 2. \(x*x=0\); 3. \((x*z)*(y*z)=z*y\); for all \(x,y,z\in X\). Unfortunately, they do not refer to a result proved by \textit{M. Kondo} and \textit{Y. B. Jun}, [Sci. Math. Jpn. 57, No. 1, 197-199 (2003; Zbl 1033.06014)] that 1. For every \(B\)-algebra \((X,*,0)\), if we define an operation \(\cdot\) by \(x\cdot y=x*(0*y)\) then \((X,\cdot,0)\) is a group and conversely, 2. For any group \((G,\cdot,e)\), if we define an operation \(*\) by \(x*y=x\cdot y^{-1}\) then \((G,*,e)\) is a \(B\)-algebra. Therefore all the results in this paper can be applied to groups.
reviewed by: Michiro Kondo
Mathematics Subject Classification ID: 06F35
Mathematics Subject Classification ID: 03G25
zbMATH DE Number: 6484530
zbMATH Keywords: finite B-algebras
zbMATH Keywords: strong BM-algebras
MaRDI profile type: Publication
full work available at URL: s13370-014-0249-8
OpenAlex ID: W2054418487
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DOI: 10.1007/S13370-014-0249-8
description: scientific article; zbMATH DE number 6484530