and right division are taken as primitive. A quasigroup is a type (2,2,2) algebra (i.e., equipped with three binary operations) satisfying the identities: y = x ∗ (x \ y), y = x \ (x ∗ y), y = (y / x) ∗ x, y = (y ∗ x) / x. In other words: Multiplication and division in either order, one after the other, on the same side by the same element, have no net effect. Hence if is a quasigroup according to the first definition, then is the same quasigroup in the sense of universal algebra. And vice versa: if is a quasigroup according to the sense of universal algebra, then is a quasigroup according to the first definition. Loops A loop is a quasigroup with an identity element; that is, an element, e, such that x ∗ e = x and e ∗ x = x for all x in Q. It follows that the identity element, e, is unique, and that every element of Q has unique left and right inverses (which need not be the same). A quasigroup with an idempotent element is called a pique ("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given an abelian group, , taking its subtraction operation as quasigroup multiplication yields a pique with the group identity (zero) turned into a "pointed idempotent". (That is, there is a principal isotopy .) A loop that is associative is a group. A group can have a non-associative pique isotope, but it cannot have a nonassociative loop isotope. There are weaker associativity properties that have been given special names. For instance, a Bol loop is a loop that satisfies either: x ∗ (y ∗ (x ∗ z)) = (x ∗ (y ∗ x)) ∗ z for each x, y and z in Q (a left Bol loop), or else ((z ∗ x) ∗ y) ∗ x = z ∗ ((x ∗ y) ∗ x) for each x, y and z in Q (a right Bol loop). A loop that is both a left and right Bol loop is a Moufang loop. This is equivalent to any one of the following single Moufang identities holding for all x, y, z: x ∗ (y ∗ (x ∗ z)) = ((x ∗ y) ∗ x) ∗ z, z ∗ (x ∗ (y ∗ x)) = ((z ∗