On analytic \(T\)-algebras (Q2740964)

In this paper, the authors give some constructions satisfying basic relations in BCK and related systems (BCH, BH, BHK, BHN, d-algebras). Consequently we have many useful and interesting examples of BCK and various related systems. To define \(x*y\) for any real numbers \(x\), \(y\), two important functions are introduced, namely NEWLINE\[NEWLINEx*y=\begin{cases} 0 & \text{if\;}x=0,\\ \frac{(x-y)^2}{x} & \text{otherwise}.\end{cases}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lambda(x,y)=\begin{cases} 0 & \text{if\;}x\leq y,\\ \frac{x-y}{x} f(x,y)>0 & \text{otherwise}.\end{cases}NEWLINE\]NEWLINE where \(f(x,y)\) is a non-negative valued function on \(\mathbb{R}\times\mathbb{R}\).NEWLINENEWLINENEWLINEThese functions satisfy basic conditions of BCK or related systems. For example, \(x * y=\text{Max}\{0,\frac{x(x-y)}{x+y}\}\) defines a d-BH algebra. \(x * y=|\frac{x-y}{x+y}|\) defines a BH-algebra, but not a d-algebra. The paper contains many useful facts.