Publication at Faculty of Mathematics and Physics |

2022

Let Q be a loop. The mappings x bar right arrow ax, x bar right arrow xa and x bar right arrow a/x are denoted by L-a, R-a. and D-a, respectively.

The loop is said to be middle Bruck if for all a, b is an element of Q there exists c is an element of Q such that DaDbDa = D-c. The right inverse of Q is the loop with operation x/(y\1).

It is proved that Q is middle Bruck if and only if the right inverse of Q is left Bruck (i.e., a left Bol loop in which (xy)(-1) = x(-1) y(-1)). Middle Bruck loops are characterized in group theoretic language as transversals T to H = G, T-G = T and t(2) = 1 for each t is an element of T.

Other results include the fact that if Q is a finite loop, then the total multiplication group is nilpotent if and only if Q is a centrally nilpotent 2-loop, and the fact that total multiplication groups of paratopic loops are isomorphic.