Semigroup–Quasigroup Structure

Identity: There exists e ∈ L e \in L e ∈ L with e a = a e = a ea = ae = a e a = a e = a for all a ∈ L a \in L a ∈ L
Divisibility: For all a , b ∈ L a, b \in L a , b ∈ L , the equations a x = b ax = b a x = b and y a = b ya = b y a = b have unique solutions

Core idea

These structures impose different conditions on a binary operation. Here, division means that each equation $ax=b$ and $ya=b$ has a unique solution for every pair $a,b$ .

Structure	Required properties
Magma	A set with a binary operation
Semigroup	An associative magma
Unital magma	A magma with a two-sided identity
Monoid	An associative magma with a two-sided identity
Quasigroup	A magma with division
Loop	A quasigroup with a two-sided identity
Group	A monoid in which every element has a two-sided inverse; equivalently, an associative loop