Universal algebra (sometimes called general algebra) is the field of

_{1},...,''x''_{''n''}). Some researchers allow

^{−1}. The axioms become:
* Associativity: .
* Identity element: ; formally: ∀''x''. ''e''∗''x''=''x''=''x''∗''e''.
* Inverse element: formally: ∀''x''. ''x''∗~''x''=''e''=~''x''∗''x''.
To summarize, the usual definition has:
* a single binary operation (

_{''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''})). (Sometimes the subscripts on ''f'' are taken off when it is clear from context which algebra the function is from.) For example, if ''e'' is a constant (nullary operation), then ''h''(''e''_{''A''}) = ''e''_{''B''}. If ~ is a unary operation, then ''h''(~''x'') = ~''h''(''x''). If ∗ is a binary operation, then ''h''(''x'' ∗ ''y'') = ''h''(''x'') ∗ ''h''(''y''). And so on. A few of the things that can be done with homomorphisms, as well as definitions of certain special kinds of homomorphisms, are listed under the entry

An Invitation to General Algebra and Universal Constructions

' (pub. Henry Helson, 15 the Crescent, Berkeley CA, 94708) 398 pp. . * Birkhoff, Garrett, 1946. Universal algebra. ''Comptes Rendus du Premier Congrès Canadien de Mathématiques'', University of Toronto Press, Toronto, pp. 310–326. * Burris, Stanley N., and H.P. Sankappanavar, 1981.

' Springer-Verlag. ''Free online edition''. * Cohn, Paul Moritz, 1981. ''Universal Algebra''. Dordrecht, Netherlands: D.Reidel Publishing. ''(First published in 1965 by Harper & Row)'' * Freese, Ralph, and Ralph McKenzie, 1987.

Commutator Theory for Congruence Modular Varieties

1st ed. London Mathematical Society Lecture Note Series, 125. Cambridge Univ. Press. . Free online second edition''. * Grätzer, George, 1968. ''Universal Algebra'' D. Van Nostrand Company, Inc. * Higgins, P. J

Groups with multiple operators

Proc. London Math. Soc. (3) 6 (1956), 366–416. * Higgins, P.J., Algebras with a scheme of operators. ''Mathematische Nachrichten'' (27) (1963) 115–132. * Hobby, David, and Ralph McKenzie, 1988.

The Structure of Finite Algebras

' American Mathematical Society. . ''Free online edition.'' * Jipsen, Peter, and Henry Rose, 1992.

', Lecture Notes in Mathematics 1533. Springer Verlag. . ''Free online edition''. * Pigozzi, Don.

General Theory of Algebras

'. ''Free online edition.'' * Smith, J.D.H., 1976. ''Mal'cev Varieties'', Springer-Verlag. * Alfred North Whitehead, Whitehead, Alfred North, 1898.

A Treatise on Universal Algebra

', Cambridge. (''Mainly of historical interest.'')

''Algebra Universalis''

a journal dedicated to Universal Algebra. {{Authority control Universal algebra,

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

that studies algebraic structure
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups
A group is a number of people or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic identi ...

as the object of study, in universal algebra one takes the class of groups as an object of study.
Basic idea

In universal algebra, an algebra (or algebraicstructure
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
...

) is a set ''A'' together with a collection of operations on ''A''. An ''n''- ary operation on ''A'' is a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

that takes ''n'' elements of ''A'' and returns a single element of ''A''. Thus, a 0-ary operation (or ''nullary operation'') can be represented simply as an element of ''A'', or a ''constant
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics)
In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...

'', often denoted by a letter like ''a''. A 1-ary operation (or ''unary operation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

'') is simply a function from ''A'' to ''A'', often denoted by a symbol placed in front of its argument, like ~''x''. A 2-ary operation (or ''binary operation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

'') is often denoted by a symbol placed between its arguments, like ''x'' ∗ ''y''. Operations of higher or unspecified ''arity
Arity () is the number of arguments
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logo ...

'' are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like ''f''(''x'',''y'',''z'') or ''f''(''x''infinitary
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

operations, such as $\backslash textstyle\backslash bigwedge\_\; x\_\backslash alpha$ where ''J'' is an infinite index set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a Set (mathematics), set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. Th ...

, thus leading into the algebraic theory of complete lattice
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s. One way of talking about an algebra, then, is by referring to it as an algebra of a certain type $\backslash Omega$, where $\backslash Omega$ is an ordered sequence of natural numbers representing the arity of the operations of the algebra.
Equations

After the operations have been specified, the nature of the algebra is further defined byaxiom
An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

s, which in universal algebra often take the form of identities, or equational laws. An example is the associative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

axiom for a binary operation, which is given by the equation ''x'' ∗ (''y'' ∗ ''z'') = (''x'' ∗ ''y'') ∗ ''z''. The axiom is intended to hold for all elements ''x'', ''y'', and ''z'' of the set ''A''.
Varieties

A collection of algebraic structures defined by identities is called avariety
Variety may refer to:
Science and technology
Mathematics
* Algebraic variety, the set of solutions of a system of polynomial equations
* Variety (universal algebra), classes of algebraic structures defined by equations in universal algebra
Hort ...

or equational class.
Restricting one's study to varieties rules out:
* quantification, including universal quantification
In mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...

($\backslash forall$) except before an equation, and existential quantification
In predicate logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order ...

($\backslash exists$)
* logical connective
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

s other than conjunction
Conjunction may refer to:
* Conjunction (astronomy), in which two astronomical bodies appear close together in the sky
* Conjunction (astrology), astrological aspect in horoscopic astrology
* Conjunction (grammar), a part of speech
* Logical conjun ...

(∧)
* relations
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...

other than equality, in particular inequalities
Inequality may refer to:
Economics
* Attention inequality
Attention inequality is a term used to target the inequality of distribution of attention across users on social networks, people in general, and for scientific papers. Yun Family Found ...

, both and order relations
The study of equational classes can be seen as a special branch of model theory
In mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...

, typically dealing with structures having operations only (i.e. the type
Type may refer to:
Science and technology Computing
* Typing, producing text via a keyboard, typewriter, etc.
* Data type, collection of values used for computations.
* File type
* TYPE (DOS command), a command to display contents of a file.
* Type ...

can have symbols for functions but not for relations
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...

other than equality), and in which the language used to talk about these structures uses equations only.
Not all algebraic structure
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s in a wider sense fall into this scope. For example, ordered group
In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order
Image:Hasse diagram of powerset of 3.svg, 250px, The Hasse diagram of the power set, set of all subsets of a three-element set , ordered by inclus ...

s involve an ordering relation, so would not fall within this scope.
The class of field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

s is not an equational class because there is no type (or "signature") in which all field laws can be written as equations (inverses of elements are defined for all ''non-zero'' elements in a field, so inversion cannot be added to the type).
One advantage of this restriction is that the structures studied in universal algebra can be defined in any category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

that has ''finite products''. For example, a topological group
In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...

is just a group in the category of topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s.
Examples

Most of the usual algebraic systems of mathematics are examples of varieties, but not always in an obvious way, since the usual definitions often involve quantification or inequalities.Groups

As an example, consider the definition of agroup
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

. Usually a group is defined in terms of a single binary operation ∗, subject to the axioms:
* Associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...

(as in the previous section): ''x'' ∗ (''y'' ∗ ''z'') = (''x'' ∗ ''y'') ∗ ''z''; formally: ∀''x'',''y'',''z''. ''x''∗(''y''∗''z'')=(''x''∗''y'')∗''z''.
* Identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

: There exists an element ''e'' such that for each element ''x'', one has ''e'' ∗ ''x'' = ''x'' = ''x'' ∗ ''e''; formally: ∃''e'' ∀''x''. ''e''∗''x''=''x''=''x''∗''e''.
* Inverse element
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

: The identity element is easily seen to be unique, and is usually denoted by ''e''. Then for each ''x'', there exists an element ''i'' such that ''x'' ∗ ''i'' = ''e'' = ''i'' ∗ ''x''; formally: ∀''x'' ∃''i''. ''x''∗''i''=''e''=''i''∗''x''.
(Some authors also use the " closure" axiom that ''x'' ∗ ''y'' belongs to ''A'' whenever ''x'' and ''y'' do, but here this is already implied by calling ∗ a binary operation.)
This definition of a group does not immediately fit the point of view of universal algebra, because the axioms of the identity element and inversion are not stated purely in terms of equational laws which hold universally "for all ..." elements, but also involve the existential quantifier "there exists ...". The group axioms can be phrased as universally quantified equations by specifying, in addition to the binary operation ∗, a nullary operation ''e'' and a unary operation ~, with ~''x'' usually written as ''x''signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a s ...

(2))
* 1 equational law (associativity)
* 2 quantified laws (identity and inverse)
while the universal algebra definition has:
* 3 operations: one binary, one unary, and one nullary (signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a s ...

(2,1,0))
* 3 equational laws (associativity, identity, and inverse)
* no quantified laws (except outermost universal quantifiers, which are allowed in varieties)
A key point is that the extra operations do not add information, but follow uniquely from the usual definition of a group. Although the usual definition did not uniquely specify the identity element ''e'', an easy exercise shows it is unique, as is each inverse element
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

.
The universal algebra point of view is well adapted to category theory. For example, when defining a group objectIn category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...

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). Further, the inverse and identity are specified as morphisms in the category. For example, in a topological group
In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...

, the inverse must not only exist element-wise, but must give a continuous mapping (a morphism). Some authors also require the identity map to be a closed inclusion (a cofibrationIn mathematics, in particular homotopy theory, a continuous mapping
:i: A \to X,
where A and X are topological spaces, is a cofibration if it lets homotopy classes of maps ,Sbe extended to homotopy classes of maps ,Swhenever a map f \in \te ...

).
Other examples

Most algebraic structures are examples of universal algebras. *Rings
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck
Ring may also refer to:
Sounds
* Ri ...

, semigroup
In mathematics, a semigroup is an algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...

s, quasigroup
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s, groupoid
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s, magmas
Magma () is the molten or semi-molten natural material from which all igneous rock
Igneous rock (derived from the Latin word ''ignis'' meaning fire), or magmatic rock, is one of the three main The three types of rocks, rock types, the other ...

, loops, and others.
* Vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s over a fixed field and modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a syst ...

over a fixed ring are universal algebras. These have a binary addition and a family of unary scalar multiplication operators, one for each element of the field or ring.
Examples of relational algebras include semilatticeIn mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (mathematics), join (a least upper bound) for any nonempty set, nonempty finite set, finite subset. Duality (order theory), Dually, a meet-semilattic ...

s, lattices, and Boolean algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s.
Basic constructions

We assume that the type, $\backslash Omega$, has been fixed. Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product. Ahomomorphism
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

between two algebras ''A'' and ''B'' is a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

''h'': ''A'' → ''B'' from the set A to the set B such that, for every operation ''f''Homomorphism
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

. In particular, we can take the homomorphic image of an algebra, ''h''(''A'').
A subalgebra of ''A'' is a subset of ''A'' that is closed under all the operations of ''A''. A product of some set of algebraic structures is the cartesian product
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of the sets with the operations defined coordinatewise.
Some basic theorems

* Theisomorphism theorems
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

, which encompass the isomorphism theorems of groups
A group is a number of people or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic identi ...

, rings
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck
Ring may also refer to:
Sounds
* Ri ...

, modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a syst ...

, etc.
* Birkhoff's HSP Theorem
In universal algebra, a variety of algebras or equational class is the class (set theory), class of all algebraic structures of a given signature (logic), signature satisfying a given set of mathematical identity#Logic and universal algebra, identi ...

, which states that a class of algebras is a variety
Variety may refer to:
Science and technology
Mathematics
* Algebraic variety, the set of solutions of a system of polynomial equations
* Variety (universal algebra), classes of algebraic structures defined by equations in universal algebra
Hort ...

if and only if it is closed under homomorphic images, subalgebras, and arbitrary direct products.
Motivations and applications

In addition to its unifying approach, universal algebra also gives deep theorems and important examples and counterexamples. It provides a useful framework for those who intend to start the study of new classes of algebras. It can enable the use of methods invented for some particular classes of algebras to other classes of algebras, by recasting the methods in terms of universal algebra (if possible), and then interpreting these as applied to other classes. It has also provided conceptual clarification; as J.D.H. Smith puts it, ''"What looks messy and complicated in a particular framework may turn out to be simple and obvious in the proper general one."'' In particular, universal algebra can be applied to the study ofmonoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

s, rings
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck
Ring may also refer to:
Sounds
* Ri ...

, and lattices. Before universal algebra came along, many theorems (most notably the isomorphism theorem
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s) 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.
The 1956 paper by Higgins referenced below has been well followed up for its framework for a range of particular algebraic systems, while his 1963 paper is notable for its discussion of algebras with operations which are only partially defined, typical examples for this being categories and groupoids. This leads on to the subject of higher-dimensional algebra which can be defined as the study of algebraic theories with partial operations whose domains are defined under geometric conditions. Notable examples of these are various forms of higher-dimensional categories and groupoids.
Constraint satisfaction problem

Universal algebra provides a natural language for the constraint satisfaction problem (CSP). CSP refers to an important class of computational problems where, given a relational algebra and an existentialsentence
Sentence(s) or The Sentence may refer to:
Common uses
* Sentence (law), the punishment a judge gives to a defendant found guilty of a crime
* Sentence (linguistics), a grammatical unit of language
* Sentence (mathematical logic), a formula not cont ...

$\backslash varphi$ over this algebra, the question is to find out whether $\backslash varphi$ can be satisfied in . The algebra is often fixed, so that refers to the problem whose instance is only the existential sentence $\backslash varphi$.
It is proved that every computational problem can be formulated as for some algebra .
For example, the problem can be stated as CSP of the algebra $\backslash big(\backslash ,\; \backslash neq\backslash big)$, i.e. an algebra with $n$ elements and a single relation, inequality.
The dichotomy conjecture (proved in April 2017) states that if is a finite algebra, then is either P or NP-complete
In computational complexity theory
Computational complexity theory focuses on classifying computational problem
In theoretical computer science
An artistic representation of a Turing machine. Turing machines are used to model general computi ...

.
Generalizations

Universal algebra has also been studied using the techniques ofcategory theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

. 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. Alternatively, one can describe algebraic structures using monad
Monad may refer to:
Philosophy
* Monad (philosophy)
Monad (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeas ...

s. The two approaches are closely related, with each having their own advantages.
In particular, every Lawvere theory gives a monad on the category of sets, while any "finitary" monad on the category of sets arises from a Lawvere theory. However, a monad describes algebraic structures within one particular category (for example the category of sets), while algebraic theories describe structure within any of a large class of categories (namely those having finite products
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produc ...

).
A more recent development in category theory is operad theory
In mathematics, an operad is concerned with prototypical algebra over a field, algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity. Operads generalize the various associativity proper ...

– 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. Thus, rings can be described as the so-called "algebras" of some operad, but not groups, since the law $g\; g^\; =\; 1$ duplicates the variable ''g'' on the left side and omits it on the right side. At first this may seem to be a troublesome restriction, but the payoff is that operads have certain advantages: for example, one can hybridize the concepts of ring and vector space to obtain the concept of associative algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, but one cannot form a similar hybrid of the concepts of group and vector space.
Another development is partial algebra where the operators can be partial function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s. Certain partial functions can also be handled by a generalization of Lawvere theories known as essentially algebraic theories.
Another generalization of universal algebra is model theory
In mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...

, which is sometimes described as "universal algebra + logic".
History

InAlfred North Whitehead
Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics ...

's book ''A Treatise on Universal Algebra,'' published in 1898, the term ''universal algebra'' had essentially the same meaning that it has today. Whitehead credits William Rowan Hamilton
Sir William Rowan Hamilton LL.D, DCL, MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician, Andrews Professor of Astronomy at Trinity College Dublin
, name_Latin = Collegium Sanctae et Individuae Trinitatis Reg ...

and Augustus De Morgan
Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmeti ...

as originators of the subject matter, and James Joseph Sylvester
James Joseph Sylvester (3 September 1814 – 15 March 1897) was an United Kingdom, English mathematician. He made fundamental contributions to Matrix (mathematics), matrix theory, invariant theory, number theory, Integer partition, partitio ...

with coining the term itself.
At the time structures such as Lie algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s and hyperbolic quaternion
In abstract algebra, the algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ...

s drew attention to the need to expand algebraic structures beyond the associatively multiplicative class. In a review Alexander Macfarlane
Prof Alexander Macfarlane FRSE LLD (21 April 1851 – 28 August 1913) was a Scottish logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argum ...

wrote: "The main idea of the work is not unification of the several methods, nor generalization of ordinary algebra so as to include them, but rather the comparative study of their several structures." At the time George Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught
Autodidacticism (also autodidactism) or self-education (also self-learning and self-teaching) is education
Education is the process of facil ...

's algebra of logic made a strong counterpoint to ordinary number algebra, so the term "universal" served to calm strained sensibilities.
Whitehead's early work sought to unify quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion a ...

(due to Hamilton), Grassmann
Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath, known in his day as a linguistics, linguist and now also as a mathematics, mathematician. He was also a physics, physicist, gener ...

's , and Boole's algebra of logic. Whitehead wrote in his book:
:''"Such algebras have an intrinsic value for separate detailed study; also they are worthy of comparative study, for the sake of the light thereby thrown on the general theory of symbolic reasoning, and on algebraic symbolism in particular. The comparative study necessarily presupposes some previous separate study, comparison being impossible without knowledge."''
Whitehead, however, had no results of a general nature. Work on the subject was minimal until the early 1930s, when Garrett Birkhoff
Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

and Øystein Ore began publishing on universal algebras. Developments in metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term it ...

and category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

in the 1940s and 1950s furthered the field, particularly the work of Abraham Robinson
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...

, Alfred Tarski
Alfred Tarski (; January 14, 1901 – October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. was a Polish-American logician ...

, Andrzej Mostowski
Andrzej Mostowski (1 November 1913 – 22 August 1975) was a Polish mathematician. He is perhaps best remembered for the Mostowski collapse lemma.
Biography
Born in Lemberg, Austria-Hungary, Mostowski entered University of Warsaw in 1931. He w ...

, and their students.
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. Although the development of mathematical logic had made applications to algebra possible, they came about slowly; results published by Anatoly MaltsevAnatoly Ivanovich Maltsev (also: Malcev, Mal'cev; Russian: Анато́лий Ива́нович Ма́льцев; 27 November N.S./14 November O.S. 1909, Moscow Governorate
Moscow Governorate (russian: Московская губерния; Rus ...

in the 1940s went unnoticed because of the war. Tarski's lecture at the 1950 International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the Nevanlinna Prize (to be rename ...

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
Leon Albert Henkin (Brooklyn, New York, April 19, 1921-Oakland, California, November 1, 2006) was one of the most important logicians and mathematicians of the 20th century. His works played a strong role in the development of logic, particularly ...

, , Roger Lyndon
Roger Conant Lyndon (December 18, 1917 – June 8, 1988) was an American mathematician, for many years a professor at the University of Michigan.. He is known for Lyndon words, the Curtis–Hedlund–Lyndon theorem, Craig interpolation, Craig–Lynd ...

, and others.
In the late 1950s, Edward Marczewski
Edward Marczewski (15 November 1907 – 17 October 1976) was a Polish mathematician. He was born Szpilrajn but changed his name while hiding from Nazi persecution.
Marczewski was a member of the Warsaw School (mathematics), Warsaw School of Mathem ...

emphasized the importance of free algebras, leading to the publication of more than 50 papers on the algebraic theory of free algebras by Marczewski himself, together with Jan Mycielski
Jan Mycielski (born February 7, 1932 in Wiśniowa, Strzyżów County, Wiśniowa, Podkarpackie Voivodeship, Poland)C ...

, Władysław Narkiewicz, Witold Nitka, J. Płonka, S. Świerczkowski, K. Urbanik, and others.
Starting with William Lawvere's thesis in 1963, techniques from category theory have become important in universal algebra.
See also

* Graph algebra * Term algebra * Clone (algebra), Clone * Universal algebraic geometry * Simple universal algebraFootnotes

References

* Bergman, George M., 1998.An Invitation to General Algebra and Universal Constructions

' (pub. Henry Helson, 15 the Crescent, Berkeley CA, 94708) 398 pp. . * Birkhoff, Garrett, 1946. Universal algebra. ''Comptes Rendus du Premier Congrès Canadien de Mathématiques'', University of Toronto Press, Toronto, pp. 310–326. * Burris, Stanley N., and H.P. Sankappanavar, 1981.

' Springer-Verlag. ''Free online edition''. * Cohn, Paul Moritz, 1981. ''Universal Algebra''. Dordrecht, Netherlands: D.Reidel Publishing. ''(First published in 1965 by Harper & Row)'' * Freese, Ralph, and Ralph McKenzie, 1987.

Commutator Theory for Congruence Modular Varieties

1st ed. London Mathematical Society Lecture Note Series, 125. Cambridge Univ. Press. . Free online second edition''. * Grätzer, George, 1968. ''Universal Algebra'' D. Van Nostrand Company, Inc. * Higgins, P. J

Groups with multiple operators

Proc. London Math. Soc. (3) 6 (1956), 366–416. * Higgins, P.J., Algebras with a scheme of operators. ''Mathematische Nachrichten'' (27) (1963) 115–132. * Hobby, David, and Ralph McKenzie, 1988.

The Structure of Finite Algebras

' American Mathematical Society. . ''Free online edition.'' * Jipsen, Peter, and Henry Rose, 1992.

', Lecture Notes in Mathematics 1533. Springer Verlag. . ''Free online edition''. * Pigozzi, Don.

General Theory of Algebras

'. ''Free online edition.'' * Smith, J.D.H., 1976. ''Mal'cev Varieties'', Springer-Verlag. * Alfred North Whitehead, Whitehead, Alfred North, 1898.

A Treatise on Universal Algebra

', Cambridge. (''Mainly of historical interest.'')

External links

''Algebra Universalis''

a journal dedicated to Universal Algebra. {{Authority control Universal algebra,