Quasigroup

A Magma ( G , ⋅ ) 𝐺 ⋅ (G,\cdot) is a quasigroup if it is a Left Quasigroup and a Right Quasigroup. We shall denote the inverses λ x : G → G : subscript 𝜆 𝑥 → 𝐺 𝐺 \lambda_{x}:G\to G and ρ x : G → G : subscript 𝜌 𝑥 → 𝐺 𝐺 \rho_{x}:G\to G , resp.

Via Latin Squares

Equivalently, a Magma ( G , ⋅ ) 𝐺 ⋅ (G,\cdot) is a quasigroup if for every x , y : G : 𝑥 𝑦 𝐺 x,y:G there exists a unique pair a , b : G : 𝑎 𝑏 𝐺 a,b:G such that a ⋅ x = y = x ⋅ b ⋅ 𝑎 𝑥 𝑦 ⋅ 𝑥 𝑏 a\cdot x=y=x\cdot b .

For the forward direction, suppose that ( G , ⋅ ) 𝐺 ⋅ (G,\cdot) is a quasigroup, and let x , y : G : 𝑥 𝑦 𝐺 x,y:G . Note that ρ x ⁢ ( y ) , λ x ⁢ ( y ) subscript 𝜌 𝑥 𝑦 subscript 𝜆 𝑥 𝑦 \rho_{x}(y),\lambda_{x}(y) validates the latin square property:

ρx(y)⋅x=((−⋅x)∘ρx)(y)=((x⋅−)∘λx)(y)=x⋅λx(y)\rho_{x}(y)\cdot x=((-\cdot x)\circ\rho_{x})(y)=((x\cdot-)\circ\lambda_{x})(y)% =x\cdot\lambda_{x}(y)

Let a , b : G : 𝑎 𝑏 𝐺 a,b:G such that a ⋅ x = y = x ⋅ b ⋅ 𝑎 𝑥 𝑦 ⋅ 𝑥 𝑏 a\cdot x=y=x\cdot b . Note that a ⋅ x = y = ρ x ⁢ ( y ) ⋅ x ⋅ 𝑎 𝑥 𝑦 ⋅ subscript 𝜌 𝑥 𝑦 𝑥 a\cdot x=y=\rho_{x}(y)\cdot x ;

Via Horn Fillers

Another motivation for quasigroups is that they obey a sort of horn filling property: namely, every 2-Horn has a (unique) filler; the lack of associativity comes from higher horns lacking fillers.

The inner horn filler is given by x ⋅ y ⋅ 𝑥 𝑦 x\cdot y ; uniqueness comes from the fact that ⋅ ⋅ \cdot is a function. Moreover, left fillers are given by λ x ⁢ ( y ) subscript 𝜆 𝑥 𝑦 \lambda_{x}(y) and right fillers by ρ x ⁢ ( y ) subscript 𝜌 𝑥 𝑦 \rho_{x}(y) .