COMMUTANTS OF MIDDLE BOL LOOPS

Abstract

The commutant of a loop is the set of all its elements that commute with each element of the loop. It is known that the commutant of a left or right Bol loop is not a subloop in general. Below we prove that the commutant of a middle Bol loop is an AIP-subloop, i.e., a subloop for which the inversion is an automorphism. A necessary and su cient condition when the commutant is invariant under the existing isostrophy between middle Bol loops and the corresponding right Bol loops is given.