New Algebraic Characterizations of Middle Bol Loop

The concept of isostrophy of a loop is a generalization of parastrophy since an isostrophy of a loop is a map (transfor-mation) that combines an isotopy (bijections) with paras-trophy. This paper examines the properties of the middle Bol loop (MBL) under the isostrophy of a loop to further introduce new algebraic characteristics of MBL. We establish a necessary and sufficient condition for some loops to be a MBL under isostrophy. It is revealed that a MBL under isostrophy of a loop has an alternative property. The necessary and sufficient conditions for a MBL under the isostrophy of a loop to be an inverse property loop were shown. We show that a middle Bol loop under isostrophy is a Steiner loop. In addition, it is demonstrated that a commutative loop under the isostrophy of MBL is Moufang. It is further obtained that commutative inverse property loops under the isostrophy of MBL are commutative Moufang loops.