Abstract algebra studies algebraic structures like groups, rings, fields and algebras.
Algebraic operation $\omega: X^n \to X$ is a mapping from the $n$-th Cartesian product $X^n$ of a set to the set $X$ itself, where $n$ is the arity of the algebraic operation: nullary operation, $n = 0$; unary operation, $n = 1$; binary operation, $n = 2$; finitary operation, $n \in \mathbb{N}$; infinitary operation, $n = \aleph_\alpha, \alpha \in \mathbb{N}$.
Algebraic system $(X, (\omega_1, \cdots), (R_1, \cdots))$ is a set $X$ with a class $(\omega_1, \cdots)$ of finitary operations and a class $(R_1, \cdots)$ of relations; such operations and relations are called basic or primitive to the algebraic system. Universal algebra or algebra is an algebraic system that has no basic relation. Relational system or model (in logic) is an algebraic system that has no basic operation.
Magma (原群), semigroup (半群), monoid (幺半群), group (群), and abelian group (交换群) are algebras $(X, ∗)$ where $∗$ is a binary operation satisfying a cumulative list of properties, defined in the following table.
Table: Group-like Algebras $(X, ∗)$ by Cumulative Properties of the Operation
Property | Property Definition | Algebra |
---|---|---|
closure | $\forall a,b \in X, a ∗ b \in X$ | magma |
associativity | $\forall a,b,c \in X, (a ∗ b) ∗ c = a ∗ (b ∗ c)$ | semigroup |
identity | $\exists e \in X, \forall a \in X, e ∗ a = a ∗ e = a$ | monoid |
inverse | $\forall a \in X, \exists b \in X, b ∗ a = a ∗ b = e$ | group |
commutativity | $\forall a,b \in X, a ∗ b = b ∗ a$ | abelian group |
Examples of groups:
Ring $(X, (+, ×))$ is a set $X$ with two binary operations called addition $+$ and multiplication $×$, such that:
For a ring $(X, (+, ×))$, zero $0$ denotes the identity element of $+$ (additive identity), and $1$ denotes the identity element of $×$ (multiplicative identity). Subtraction $-$ is the inverse operation of addition $+$.
Basic properties:
Examples of rings:
Semiring is an algebra similar to a ring, but without additive inverse elements. Commutative ring is a ring whose multiplication is commutative. Division ring is a ring with multiplicative inverses for all nonzero elements; division is the inverse operation of multiplication $+$, defined for all nonzero elements of a division ring.
Field $(X, (+, ×))$ is a commutative division ring.
Examples of fields: finite field with four elements; the fields of rational numbers $(\mathbb{Q}, (+,×))$, real numbers $(\mathbb{R}, (+,×))$, and complex numbers $(\mathbb{C}, (+,×))$.
Unary algebra.
Algebraic lattice. Boolean algebra.
Banach algebra.