Equational axioms

An axiom of an algebraic structure often has the form of an identity, that is, an equation such that the two sides of the equals sign are expressions that involve operations of the algebraic structure and variables. If the variables in the identity are replaced by arbitrary elements of the algebraic structure, the equality must remain true. Here are some common examples.

Commutativity

An operation ${\displaystyle *}$ is commutative if

$${\displaystyle x*y=y*x}$$

for every x and y in the algebraic structure.

Associativity

An operation ${\displaystyle *}$ is associative if

$${\displaystyle (x*y)*z=x*(y*z)}$$

for every x, y and z in the algebraic structure.

Left distributivity

An operation ${\displaystyle *}$ is left distributive with respect to another operation ${\displaystyle +}$ if

$${\displaystyle x*(y+z)=(x*y)+(x*z)}$$

for every x, y and z in the algebraic structure (the second operation is denoted here as +, because the second operation is addition in many common examples).

Right distributivity

An operation ${\displaystyle *}$ is right distributive with respect to another operation ${\displaystyle +}$ if

$${\displaystyle (y+z)*x=(y*x)+(z*x)}$$

for every x, y and z in the algebraic structure.

Distributivity

An operation ${\displaystyle *}$ is distributive with respect to another operation ${\displaystyle +}$ if it is both left distributive and right distributive. If the operation ${\displaystyle *}$ is commutative, left and right distributivity are both equivalent to distributivity.

Existential axioms

Some common axioms contain an existential clause. In general, such a clause can be avoided by introducing further operations, and replacing the existential clause by an identity involving the new operation. More precisely, let us consider an axiom of the form "for all X there is y such that ${\displaystyle f(X,y)=g(X,y)}$", where X is a k-tuple of variables. Choosing a specific value of y for each value of X defines a function ${\displaystyle \varphi :X\mapsto y,}$ which can be viewed as an operation of arity k, and the axiom becomes the identity ${\displaystyle f(X,\varphi (X))=g(X,\varphi (X)).}$

The introduction of such auxiliary operation complicates slightly the statement of an axiom, but has some advantages. Given a specific algebraic structure, the proof that an existential axiom is satisfied consists generally of the definition of the auxiliary function, completed with straightforward verifications. Also, when computing in an algebraic structure, one generally uses explicitly the auxiliary operations. For example, in the case of numbers, the additive inverse is provided by the unary minus operation ${\displaystyle x\mapsto -x.}$

Also, in universal algebra, a variety is a class of algebraic structures that share the same operations, and the same axioms, with the condition that all axioms are identities. What precedes shows that existential axioms of the above form are accepted in the definition of a variety.

Here are some of the most common existential axioms.

Identity element

A binary operation ${\displaystyle *}$ has an identity element if there is an element e such that

$${\displaystyle x*e=x\quad {\text{and}}\quad e*x=x}$$

for all x in the structure. Here, the auxiliary operation is the operation of arity zero that has e as its result.

Inverse element

Given a binary operation ${\displaystyle *}$ that has an identity element e, an element x is invertible if it has an inverse element, that is, if there exists an element ${\displaystyle \operatorname {inv} (x)}$ such that

$${\displaystyle \operatorname {inv} (x)*x=e\quad {\text{and}}\quad x*\operatorname {inv} (x)=e.}$$

For example, a group is an algebraic structure with a binary operation that is associative, has an identity element, and for which all elements are invertible.

Non-equational axioms

The axioms of an algebraic structure can be any first-order formula, that is a formula involving logical connectives (such as "and", "or" and "not"), and logical quantifiers (${\displaystyle \forall ,\exists }$) that apply to elements (not to subsets) of the structure.

Such a typical axiom is inversion in fields. This axiom cannot be reduced to axioms of preceding types. (it follows that fields do not form a variety in the sense of universal algebra.) It can be stated: "Every nonzero element of a field is invertible;" or, equivalently: the structure has a unary operation inv such that

${\displaystyle \forall x,\quad x=0\quad {\text{or}}\quad x\cdot \operatorname {inv} (x)=1.}$

The operation inv can be viewed either as a partial operation that is not defined for x = 0; or as an ordinary function whose value at 0 is arbitrary and must not be used.

One set with operations

Simple structures: no binary operation:

• Set: a degenerate algebraic structure S having no operations.

Group-like structures: one binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers.

• Group: a monoid with a unary operation (inverse), giving rise to inverse elements.
• Abelian group: a group whose binary operation is commutative.

Ring-like structures or Ringoids: two binary operations, often called addition and multiplication, with multiplication distributing over addition.

• Ring: a semiring whose additive monoid is an abelian group.
• Division ring: a nontrivial ring in which division by nonzero elements is defined.
• Commutative ring: a ring in which the multiplication operation is commutative.
• Field: a commutative division ring (i.e. a commutative ring which contains a multiplicative inverse for every nonzero element).

Lattice structures: two or more binary operations, including operations called meet and join, connected by the absorption law.^[2]

• Complete lattice: a lattice in which arbitrary meet and joins exist.
• Bounded lattice: a lattice with a greatest element and least element.
• Distributive lattice: a lattice in which each of meet and join distributes over the other. A power set under union and intersection forms a distributive lattice.
• Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.

Two sets with operations

• Module: an abelian group M and a ring R acting as operators on M. The members of R are sometimes called scalars, and the binary operation of scalar multiplication is a function R × M → M, which satisfies several axioms. Counting the ring operations these systems have at least three operations.
• Vector space: a module where the ring R is a division ring or field.

• Algebra over a field: a module over a field, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and linearity with respect to multiplication.
• Inner product space: an F vector space V with a definite bilinear form V × V → F.