Universal algebra

algebraequational theoryequational reasoningalgebraicequational theoriesalgebraic theoryalgebrasdefinition of a groupgeneral algebrahomomorphic image
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.wikipedia
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Structure (mathematical logic)

structuremodelstructures
In universal algebra, an algebra (or algebraic structure) is a set A together with a collection of operations on A.
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.

Algebraic structure

algebraic structuresunderlying setalgebraic system
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
In the context of universal algebra, the set A with this structure is called an algebra, while, in other contexts, it is (somewhat ambiguously) called an algebraic structure, the term algebra being reserved for specific algebraic structures that are vector spaces over a field or modules over a commutative ring.

Outline of algebraic structures

boundary algebraList of algebraic structuresalgebra of a certain type
One way of talking about an algebra, then, is by referring to it as an algebra of a certain type \Omega, where \Omega is an ordered sequence of natural numbers representing the arity of the operations of the algebra.
Another branch of mathematics known as universal algebra studies algebraic structures in general.

Variety (universal algebra)

varietyvarietiesBirkhoff's HSP theorem
A collection of algebraic structures defined by identities is called a variety or equational class.
In the mathematical subject of universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities.

Complete lattice

completecomplete latticesarbitrary meets and joins
Some researchers allow infinitary operations, such as where J is an infinite index set, thus leading into the algebraic theory of complete lattices.
Being a special instance of lattices, they are studied both in order theory and universal algebra.

Model theory

modelmodelsmodel-theoretic
The study of equational classes can be seen as a special branch of model theory, typically dealing with structures having operations only (i.e. the type can have symbols for functions but not for relations other than equality), and in which the language used to talk about these structures uses equations only.
:universal algebra + logic = model theory.

Signature (logic)

signaturesignatureslanguage
The study of equational classes can be seen as a special branch of model theory, typically dealing with structures having operations only (i.e. the type can have symbols for functions but not for relations other than equality), and in which the language used to talk about these structures uses equations only.
In universal algebra, a signature lists the operations that characterize an algebraic structure.

Binary operation

binary operatoroperationbinary
A 2-ary operation (or binary operation) is often denoted by a symbol placed between its arguments, like x ∗ y.
However, both in universal algebra and model theory the binary operations considered are defined on all of

Group (mathematics)

groupgroupsgroup operation
As an example, consider the definition of a group.
This formulation exhibits groups as a variety of universal algebra.

Lattice (order)

latticelattice theorylattices
Examples of relational algebras include semilattices, lattices, and Boolean algebras.
Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra.

Group object

For example, when defining a group object in category theory, where the object in question may not be a set, one must use equational laws (which make sense in general categories), rather than quantified laws (which refer to individual elements).

Quasigroup

loopLoop (algebra)loops
One defines a quasigroup as a set with one binary operation, and the other, from universal algebra, defines a quasigroup as having three primitive operations.

Isomorphism theorems

first isomorphism theoremisomorphism theoremNoether isomorphism theorem
Before universal algebra came along, many theorems (most notably the isomorphism theorems) were proved separately in all of these classes, but with universal algebra, they can be proven once and for all for every kind of algebraic system.
In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

Order theory

orderorder relationordering
However, one can go even further: if all finite non-empty infima exist, then ∧ can be viewed as a total binary operation in the sense of universal algebra.

Category theory

categorycategoricalcategories
One advantage of this restriction is that the structures studied in universal algebra can be defined in any category that has finite products.
General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later; it is now applied throughout mathematics.

Operad

operadsoperad theoryassociative operad
A more recent development in category theory is operad theory – an operad is a set of operations, similar to a universal algebra, but restricted in that equations are only allowed between expressions with the variables, with no duplication or omission of variables allowed.
They form a category-theoretic analog of universal algebra.

Alfred North Whitehead

WhiteheadA. N. WhiteheadA.N. Whitehead
In Alfred North Whitehead's book A Treatise on Universal Algebra, published in 1898, the term universal algebra had essentially the same meaning that it has today.
In A Treatise on Universal Algebra (1898) the term universal algebra had essentially the same meaning that it has today: the study of algebraic structures themselves, rather than examples ("models") of algebraic structures.

Lawvere theory

Lawvere theoriesFermat theory
In this approach, instead of writing a list of operations and equations obeyed by those operations, one can describe an algebraic structure using categories of a special sort, known as Lawvere theories or more generally algebraic theories.
In category theory, a Lawvere theory (named after American mathematician William Lawvere) is a category that can be considered a categorical counterpart of the notion of an equational theory.

Partial algebra

Another development is partial algebra where the operators can be partial functions.
In abstract algebra, a partial algebra is a generalization of universal algebra to partial operations.

Free object

freefree functorfree algebra
In the period between 1935 and 1950, most papers were written along the lines suggested by Birkhoff's papers, dealing with free algebras, congruence and subalgebra lattices, and homomorphism theorems.
It is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations).

Homomorphism

homomorphichomomorphismse-free homomorphism
A homomorphism between two algebras A and B is a function h: A → B from the set A to the set B such that, for every operation f A of A and corresponding f B of B (of arity, say, n), h(f A (x 1,...,x n )) = f B (h(x 1 ),...,h(x n )).
More precisely, they are equivalent for fields, for which every homomorphism is a monomorphism, and for varieties of universal algebra, that is algebraic structures for which operations and axioms (identities) are defined without any restriction (fields are not a variety, as the multiplicative inverse is defined either as a unary operation or as a property of the multiplication, which are, in both cases, defined only for nonzero elements).

Bjarni Jónsson

JónssonJónsson, Bjarni
Tarski's lecture at the 1950 International Congress of Mathematicians in Cambridge ushered in a new period in which model-theoretic aspects were developed, mainly by Tarski himself, as well as C.C. Chang, Leon Henkin, Bjarni Jónsson, Roger Lyndon, and others.
Bjarni Jónsson (February 15, 1920 – September 30, 2016) was an Icelandic mathematician and logician working in universal algebra, lattice theory, model theory and set theory.

Term algebra

Herbrand UniverseTermsalgebraic terms
In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature.

Garrett Birkhoff

BirkhoffBirkhoff, GarrettG. Birkhoff
Work on the subject was minimal until the early 1930s, when Garrett Birkhoff and Øystein Ore began publishing on universal algebras.
His 1935 paper, "On the Structure of Abstract Algebras" founded a new branch of mathematics, universal algebra.

Clone (algebra)

cloneclonesfinitary multiple composition
Given an algebra in a signature σ, the set of operations on its carrier definable by a σ-term (the term functions) is a clone.