x) ∗ y) ∗ x, (x ∗ y) ∗ (z ∗ x) = x ∗ ((y ∗ z) ∗ x), or (x ∗ y) ∗ (z ∗ x) = (x ∗ (y ∗ z)) ∗ x. Symmetries names the following important properties and subclasses: Semisymmetry A quasigroup is semisymmetric if the following equivalent identities hold: x ∗ y = y / x, y ∗ x = x \ y, x = (y ∗ x) ∗ y, x = y ∗ (x ∗ y). Although this class may seem special, every quasigroup Q induces a semisymmetric quasigroup QΔ on the direct product cube Q3 via the following operation: where "//" and "\\" are the conjugate division operations given by and . Triality Total symmetry A narrower class is a totally symmetric quasigroup (sometimes abbreviated TS-quasigroup) in which all conjugates coincide as one operation: . Another way to define (the same notion of) totally symmetric quasigroup is as a semisymmetric quasigroup which also is commutative, i.e. . Idempotent total symmetric quasigroups are precisely (i.e. in a bijection with) Steiner triples, so such a quasigroup is also called a Steiner quasigroup, and sometimes the latter is even abbreviated as squag. The term sloop refers to an analogue for loops, namely, totally symmetric loops that satisfy instead of . Without idempotency, total symmetric quasigroups correspond to the geometric notion of extended Steiner triple, also called Generalized Elliptic Cubic Curve (GECC). Total antisymmetry A quasigroup is called totally anti-symmetric if for all , both of the following implications hold: (c ∗ x) ∗ y = (c ∗ y) ∗ x implies that x = y x ∗ y = y ∗ x implies that x = y. It is called weakly totally anti-symmetric if only the first implication holds. This property is required, for example, in the Damm algorithm. Examples Every group is a loop, because if and only if , and if and only if . The integers Z (or the rationals Q or the reals R) with subtraction (−) form a quasigroup. These quasiqroups are not loops because there is no identity element (0 is a right identity because , but not a left identity because, in general, ). The nonzero rationals Q× (or the nonzero reals R×) with division (÷) form a quasigroup. Any vector space over a field of characteristic not equal to 2 forms an idempotent, commutative quasigroup under the