On finite \(B\)-algebra.
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.
scientific article; zbMATH DE number 6484530
scientific article