satisfies the identities:

A left-quasigroup is a type (2, 2) algebra that satisfies the identities:

A quasigroup is a type (2, 2, 2) algebra (i.e., equipped with three binary operations) that satisfies the identities:

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 definition of the previous section, then is the same quasigroup in the sense of universal algebra.