Module un_algebra::quasigroup::quasigroup

Algebraic quasigroups.

An algebraic quasigroup is a multiplicative magma M, equipped with left and right division operators \ and /. The magma multiplication operator is not required to be associative. Both division operators must obey cancellation axioms.

Quasigroups can be defined in terms of one binary operation with the latin square property, but in un_algebra quasigroups are defined with separate left and right division operators and axioms.

Axioms

∀x, y ∈ M

Left cancellation:  x \ (x × y) = y = x × (x \ y).
Right cancellation: (x / y) × y = x = (x × y) / y.

References

See references for a formal definition of a quasigroup.

Traits

Quasigroup

An algebraic quasigroup.

Functions

left_lcancellation	The left axiom of left-cancellation.
left_rcancellation	The left axiom of right-cancellation.
num_left_lcancellation	The left axiom of numeric left-cancellation.
num_left_rcancellation	The left axiom of numeric right-cancellation.
num_right_lcancellation	The right axiom of numeric left-cancellation.
num_right_rcancellation	The right axiom of numeric right-cancellation.
right_lcancellation	The right axiom of left-cancellation.
right_rcancellation	The right axiom of right-cancellation.