A loop (Q,·,,/) is called a middle Bol loop if it obeys the identity x(yzx) = (x/z)(yx). In this paper, some new algebraic properties of a middle Bol loop are established. Four bi-variate mappings f i ,g i , i = 1,2 and four j-variate mappings α j ,β j ,φ j ,ψ j , j ∈ N are introduced and some interesting properties of the former are found. Neccessary and sufficient conditons in terms of f i ,g i , i = 1,2, for a middle Bol loop to have the elasticity property, RIP, LIP, right alternative property (RAP) and left alternative property (LAP) are establsihed. Also, neccessary and sufficient conditons in terms of α j ,β j ,φ j ,ψ j , j ∈ N, foramiddleBollooptohavepowerRAPandpower LAPareestablsihed. Neccessary and sufficient conditons in terms of f i ,g i , i = 1,2 and α j ,β j ,φ j ,ψ j , j ∈ N, for a middle Bol loop to be a group, Moufang loop or extra loop are established. A middle Bol loop is shown to belong to some classes of loops whose identiites are of the J.D. Phillips’ RIF- loop and WRIF-loop (generalizations of Moufang and Steiner loops) and WIP power associative conjugacy closed loop types if and only if some identities defined by g 1 and g 2 are obeyed.
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