\(\nu\)-regular Menger algebras (Q1272146)

A Menger algebra of rank \(n\) is an algebra of the form \((G,\cdot)\), where \(\cdot\colon(x,y_1,\ldots,y_n)\to x[y_1\ldots y_n]\) is an \((n+1)\)-ary operation on \(G\) satisfying the superassociativity law \(x[y_1\ldots y_n][z_1\ldots z_n]=x[y_1[z_1\ldots z_n]\ldots y_n[z_1\ldots z_n]]\), where \(x[y_1\ldots y_n][z_1\ldots z_n]=(x[y_1\ldots y_n])[z_1\ldots z_n]\). They are a natural generalization of semigroups. \((G,\cdot)\) is called \(\nu\)-regular if for every vector \((g_1,\ldots,g_n)\in G^n\) there exists some \(x\in G\) such that the equalities \(g_i[x\ldots x][g_1\ldots g_n]=g_i\) are true for each \(i=1,\ldots,n\). In the present paper some properties of \(\nu\)-regular Menger algebras concerning idempotents and inverse elements similar to properties known for semigroups are proved.