Remarks on \(*\)-associative groupoids (Q2759830)

In this paper there is defined an involutive groupoid as a set \(A\) with a binary operation \(\cdot\) and an unary operator \(*\) satisfying: \((x^*)^*=x\), \((x\cdot y)^*=y^*\cdot x^*\). An involutive groupoid \((A;\cdot,*)\) is called \(*\)-associated if \((x\cdot y)^*\cdot z=x\cdot(y\cdot z)^*\).NEWLINENEWLINENEWLINEFurther, the authors -- define an ideal of \(A\), a filter, a \(*\)-idempotent, a \(*\)-associative quasigroup, a poset; -- investigate the general theory of \(*\)-associative groupoids analogous with classical group theory and -- show some relations with semilattices and quasigroups. Some examples are given as an illustration of this new theory.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00014].