Set
1. There exist members
Category
1. There exist members, where each member has a map or morphism to another member. A → B, where A and B are members
n-Category
1. There exist members, where each member has a map or morphism to another member. A → B, where A and B are members
2. There are 2-morphisms between morphisms, 3-morphisms between 2-morphisms,....,n-morphisms between (n - 1)-morphisms.
Magma - In the past, this was called a groupoid, but today, we use the word "groupoid" to mean a group where the binary operation is partially defined.
1. There exists a binary operation AB = C, where A, B, and C are members
Semigroup - A semigroup is a magma that's associative.
1. There exists a binary operation AB = C, where A, B, and C are members
2. Associative A(BC) = (AB)C
Monoid - A monoid is a semigroup with an identity.
1. There exists a binary operation AB = C, where A, B, and C are members
2. Associative A(BC) = (AB)C
3. There exists an identity, where AI = IA = A
Quasigroup - A quasigroup is a group that's not associative.
1. There exists a binary operation AB = C, where A, B, and C are members
2. There exists an identity, where AI = IA = A
3. There exists an inverse for each element where AA-1 = I
Group
1. There exists a binary operation AB = C, where A, B, and C are members
2. Associative A(BC) = (AB)C
3. There exists an identity, where AI = IA = A
4. There exists an inverse for each element where AA-1 = I
Abelian Group
1. There exists a binary operation AB = C, where A, B, and C are members
2. Associative A(BC) = (AB)C
3. There exists an identity, where AI = IA = A
4. There exists an inverse for each element where AA-1 = I
5. Commutative AB = BA
Abelian Monoid
1. There exists a binary operation AB = C, where A, B, and C are members
2. Associative A(BC) = (AB)C
3. There exists an identity, where AI = IA = A
4. Commutative AB = BA
Ring - A ring is an abelian group under addition, and a semigroup under multiplication.
1. There exist two binary operations, usually called addition, +, and multiplication, x
2. Additive Associative (A + B) + C = A + (B + C)
3. Additive Identity 0 + A = A + 0 = A
4. Additive Inverse A + (-A) = (-A) + A = 0
5. Additive Commutative A + B = B + A
6. Multiplicative Associative (A x B) x C = A x (B x C)
7. Left and Right Distributive A x (B + C) = (A x B) + (A x C), (B + C) x A = (B x A) + (C x A)
Rig - A rig is a group under both addition and multiplication.
1. There exist two binary operations, usually called addition, +, and multiplication, x
2. Additive Associative (A + B) + C = A + (B + C)
3. Additive Identity 0 + A = A + 0 = A
4. Additive Inverse A + (-A) = (-A) + A = 0
5. Multiplicative Associative (A x B) x C = A x (B x C)
6. Multiplicative Identity 1 x A = A x 1 = A
7. Multiplicative Inverse A x (1/A) = (1/A) x A = 1
8. Left and Right Distributive A x (B + C) = (A x B) + (A x C), (B + C) x A = (B x A) + (C x A)
Field - A field is a rig with the additional axiom that you may not divide by zero.
1. There exist two binary operations, usually called addition, +, and multiplication, x
2. Additive Associative (A + B) + C = A + (B + C)
3. Additive Identity 0 + A = A + 0 = A
4. Additive Inverse A + (-A) = (-A) + A = 0
5. Multiplicative Associative (A x B) x C = A x (B x C)
6. Multiplicative Identity 1 x A = A x 1 = A
7. Multiplicative Inverse A x (1/A) = (1/A) x A = 1
8. Left and Right Distributive A x (B + C) = (A x B) + (A x C), (B + C) x A = (B x A) + (C x A)
9. You may not divide by 0.
A ring is an algebraic gadget such that you can add and multiply its elements consistently. A module over a ring is an algebraic gadget such that you can add its elements, but probably not multiply them, and you can multiply the elements of the module by an element of the ring, getting a new element of the module. The ring acts on the module.
Ring: the integers
Modules over this ring: additive groups
Ring: a field such as the real, complex or rational numbers
Modules over this ring: vector spaces over the field
Ring: n x n matrices
Modules over this ring: vectors with n-components
Ring: the group algebra of a group (functions on the group, convolution
product)
Modules over this ring: representations of the group
Ring: real or complex valued functions on a manifold
Modules over this ring: space of sections of a real or complex vector
bundle over the manifold
There is also a thing called a torsor which is a group that has "lost its identity".