Today we continue with Mathematics, and more specifically the branch of "Discrete Mathematics", in order to get into Group-like Structures. This is part 2, you can find part 1 here.
So, without further ado, let's get straight into it!
Magma
Easing the requirements of the definition of groups to the point that only the closure property has to be satisfied, yields an algebraic structure known as a Magma. A magma is thus a non-empty set together with a binary operation that satisfies closure (basically any function).
A magma clearly has way to few properties (only one), so let's add one additional property to come up with other structures...
Associative Magma (or Semigroup)
A magma that also satisfies associativity is basically a semigroup.
Unital Magma
A magma that satisfies identity is called an Unital magma.
Quasigroup
A magma which satisfies invertibility (or division), which basically means that division is always possible between any pair of elements is called a Quasigroup. A quasigroup doesn't have to be associative, nor does it have to include an identity element.
Loop
Adding an identity element to a Quasigroup yields a structure known as a Loop, which satisfies closure, identity and division.
Groupoid
A groupoid is commonly defined as a structure that doesn't satisfy closure but satisfies all other properties of a group (associativity, identity and invertibility). A groupoid basically easies the requirement of the binary operation, so that it doesn't have to be defined for all elements in the set.
Semigroupoid
A semigroupoid is a structure that satisfies only associativity.
Small category
A small category is a structure that satisfies associativity and has an identity element.
Partial Order Relations and Sets → Partial Order Relations, POSET (Elements, Max-Min, Upper-Lower Bounds), Hasse Diagrams, Total Order Relations, Lattices
