Overview of Algebras

                 I   R   T   A   C   D

Algebraic structure	id	rev	tot	ass	com	dis	O IRTA CD (CarOtiD)
Commutative group	i	r	t	a	c		1 irta c
Group	i	r	t	a			1 irta
Monoid	i		t	a			1 i.ta
Semigroup			t	a			1 ..ta
Magma			t				1 ..t.
--------------------	---	---	---	---	---	---	----------
Inverse Semigroup		r	t	a			1 .rta
Loop	i	r	t				1 irt.
Quasigroup		r	t				1 .rt.
--------------------	---	---	---	---	---	---	----------
Groupoid	i	r		a			1 ir.a
Semigroupoid				a			1 ...a
--------------------	---	---	---	---	---	---	----------
Category				a			1 (composition)
--------------------	---	---	---	---	---	---	----------
Ring	i	r	t	a	c	d	2 (ext_add, ext_mult)
--------------------	---	---	---	---	---	---	----------
Field	i	r	t	a	c	d	4 {+ - ÷ ×}

CAROTID

• Commutativity : C, com, comm

• Associativity : A, ass, assoc

• Invertibility : R, inv, rev, reversibility

• Number of ops : O, nr. of binary operations

• Totality : T, closure

• Identity : I, id, identity element = e

• Distributivity: D, dis, distr

Monoid: {Σ, ϵ, (* -> *)}

Groupoid

Groupoid (Brandt groupoi, virtual group) generalizes the notion of group in several equivalent ways. A groupoid can be seen as:

• Group with a partial function replacing the binary operation

• Category in which every morphism is invertible.

Field

A field is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics.

Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and Hans Hahn.

Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups.

Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure.

Field

• is a set

• on which addition, subtraction, multiplication, division are defined

• and behave as the corresponding operations on rational and real numbers

• field is a fundamental algebraic structure

Ring

a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.

Algebra over a field

an algebra (algebra over a field) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure, which consists of a set, together with operations of multiplication, addition, and scalar multiplication by elements of the underlying field, and satisfies the axioms implied by "vector space" and "bilinear".

Index of Algebraic Structures

• Group-like

  • Group

  • Abelian group (commutative group)

  • Semigroup (group without closure)

  • Monoid

  • Rack and quandle

  • Quasigroup and loop

  • Magma

  • Lie group

• Ring-like

  • Ring

  • Semiring

  • Near-ring

  • Commutative ring

  • Integral domain

  • Field Division ring

• Lattice-like

  • Lattice

  • Semilattice

  • Complemented lattice

  • Total order

  • Heyting algebra

  • Boolean algebra

• Module-like

  • Module Group with operators

  • Vector space

  • Linear algebra

• Algebra-like

  • Algebra

  • Associative

  • Non-associative

  • Composition algebra

  • Lie algebra

  • Graded

  • Bialgebra

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