Universal algebra (sometimes called general algebra) is the field of mathematics that studies the ideas common to all algebraic structures. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations,

Contents 1 Basic idea 2 Examples 2.1 Groups 3 Basic constructions 4 Some basic theorems 5 Motivations and applications 6 History 7 See also 8 Footnotes 9 References 10 External links

Basic idea

From the point of view of universal algebra, an algebra (or algebraic structure) is a set A together with a collection of operations on A. An n-ary operation on A is a function that takes n elements of A and returns a single element of A. In Logic, Mathematics, and Computer science, the arity (synonyms include type, adicity, and rank) of a function In its simplest meaning in Mathematics and Logic, an operation is an action or procedure which produces a new value from one or more input values The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Thus, a 0-ary operation (or nullary operation) can be represented simply as an element of A, or a constant, often denoted by a letter like a. A 1-ary operation (or unary operation) is simply a function from A to A, often denoted by a symbol placed in front of its argument, like ~x. In Mathematics, a unary operation is an operation with only one Operand, i A 2-ary operation (or binary operation) is often denoted by a symbol placed between its arguments, like x * y. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two Operations of higher or unspecified arity are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like f(x,y,z) or f(x₁,. . . ,x_n). Some researchers allow infinitary operations, such as where J is an infinite index set, thus leading into the algebraic theory of complete lattices. In Mathematics or Logic, a finitary operation is one like those of Arithmetic, that takes a finite number of input values to produce an output In Mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index In Mathematics, a complete lattice is a Partially ordered set in which all subsets have both a Supremum (join and an Infimum (meet One way of talking about an algebra, then, is by referring to it as an algebra of a certain type Ω, where Ω is an ordered sequence of natural numbers representing the arity of the operations of the algebra.

After the operations have been specified, the nature of the algebra can be further limited by axioms, which in universal algebra often take the form of equational laws. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject An example is the associative axiom for a binary operation, which is given by the equation x * (y * z) = (x * y) * z. In Mathematics, associativity is a property that a Binary operation can have The axiom is intended to hold for all elements x, y, and z of the set A.

Universal algebra can be seen as a special branch of model theory, in which we are typically dealing with structures having operations only (i. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models 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 Logic, especially Mathematical logic, a signature lists and describes the Non-logical symbols of a Formal language. This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations Not all algebraic structures in a wider sense fall into this scope. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, For example ordered groups are not studied in mainstream universal algebra because they involve a binary relation. In Abstract algebra, an ordered group is a group (G+ equipped with a Partial order "≤" which is translation-invariant A more fundamental restriction is that universal algebra cannot study the class of fields, because there is no type in which all field laws can be written as equations. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division One advantage of this restriction is that the structures studied in universal algebra can be defined in any category which has finite products. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as

Examples

Most of the usual algebraic systems of mathematics are examples of universal algebras, but not always in an obvious way.

Groups

To see how this works, let's consider the definition of a group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Normally a group is defined in terms of a single binary operation *, subject to these axioms:

• Associativity (as in the previous section): x * (y * z)  =  (x * y) * z. In Mathematics, associativity is a property that a Binary operation can have
• Identity element: There exists an element e such that e * x  =  x  =  x * e. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that
• Inverse element: For each x, there exists an element i such that x * i  =  e  =  i * x. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to

(Sometimes you will also see an axiom called "closure", stating that x * y belongs to the set A whenever x and y do. But from a universal algebraist's point of view, that is already implied when you call * a binary operation. )

Now, this definition of a group is problematic from the point of view of universal algebra. The reason is that the axioms of the identity element and inversion are not stated purely in terms of equational laws but also have clauses involving the phrase "there exists . . . such that . . . ". This is inconvenient; the list of group properties can be simplified to universally quantified equations if we add a nullary operation e and a unary operation ~ in addition to the binary operation *, then list the axioms for these three operations as follows:

• Associativity: x * (y * z)  =  (x * y) * z.
• Identity element: e * x  =  x  =  x * e.
• Inverse element: x * (~x)  =  e  =  (~x) * x.

(Of course, we usually write "x ^-1" instead of "~x", which shows that the notation for operations of low arity is not always as given in the second paragraph. In Logic, Mathematics, and Computer science, the arity (synonyms include type, adicity, and rank) of a function )

It's important to check that this really does capture the definition of a group. The reason that it might not is that specifying one of these universal groups might give more information than specifying one of the usual kind of group. After all, nothing in the usual definition said that the identity element e was unique; if there is another identity element e', then it's ambiguous which one should be the value of the nullary operator e. However, this is not a problem, because identity elements can be proved to be always unique. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that The same thing is true of inverse elements. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to So the universal algebraist's definition of a group really is equivalent to the usual definition.

Basic constructions

We assume that the type, Ω, has been fixed. Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product.

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 (of arity, say, n), h(f_A(x₁,. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function . . ,x_n)) = f_B(h(x₁),. . . ,h(x_n)). (Here, subscripts are placed on f to indicate whether it is the version of f in A or B. In theory, you could tell this from the context, so these subscripts are usually left off. ) 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 Homomorphism. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector 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 of the sets with the operations defined coordinatewise. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory.

Some basic theorems

• The Isomorphism theorems, which encompass the Isomorphism theorems of groups, rings, modules, etc. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars
• Birkhoff's HSP Theorem, which states that a class of algebras is a variety if and only if it is closed under homomorphic images, subalgebras, and arbitrary direct products. In Universal algebra, a variety of algebras is the class of all Algebraic structures of a given signature satisfying a given set of identities In Universal algebra, a variety of algebras is the class of all Algebraic structures of a given signature satisfying a given set of identities

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 method 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 of monoids, rings, and lattices. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' Before universal algebra came along, many theorems (most notably the isomorphism theorems) were proved separately in all of these fields, but with universal algebra, they can be proven once and for all for every kind of algebraic system. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural

A more generalised programme along these lines is carried out by category theory. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a category. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Category theory applies to many situations where universal algebra does not, extending the reach of the theorems. Conversely, many theorems that hold in universal algebra do not generalise all the way to category theory. Thus both fields of study are useful.

History

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. Alfred North Whitehead, OM ( February 15 1861, Ramsgate, Kent, England &ndash December 30 1947, Whitehead credits William Rowan Hamilton and Augustus De Morgan as originators of the subject matter, and James Joseph Sylvester with coining the term itself^[1]. Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Augustus De Morgan ( 27 June, 1806 &ndash 18 March, 1871) was a British Mathematician and Logician. James Joseph Sylvester ( September 3, 1814 London – March 15, 1897 Oxford) was an English Mathematician

At the time structures such as Lie algebras and hyperbolic quaternions drew attention to the need to expand algebraic structures beyond the associatively multiplicative class. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, a hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association In a review Alexander MacFarlane 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. Alexander Macfarlane ( April 21 1851 – August 28, 1913) was a Scottish - Canadian Logician Physicist " At the time George Boole's algebra of logic made a strong counterpoint to ordinary number algebra, so the term "universal" served to calm strained sensibilities. George Boole (buːl ( November 2, 1815 &ndash December 8, 1864) was a British Mathematician and Philosopher.

Whitehead's early work sought to unify quaternions (due to Hamilton), Grassmann's Ausdehnungslehre, and Boole's algebra of logic. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Hermann Günther Grassmann ( April 15, 1809, Stettin ( Szczecin) &ndash September 26, 1877, Stettin) was a 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. "^[2]

Whitehead, however, had no results of a general nature. Work on the subject was minimal until the early 1930s, when Garrett Birkhoff and Øystein Ore began publishing on universal algebras. Garrett Birkhoff ( January 19, 1911, Princeton, New Jersey, USA – November Øystein Ore ( 7 October 1899 in Oslo, Norway &ndash 13 August 1968 in Oslo was a Norwegian Mathematician Developments in metamathematics and category theory in the 1940s and 1950s furthered the field, particularly the work of Abraham Robinson, Alfred Tarski, Andrzej Mostowski, and their students (Brainerd 1967). In general metamathematics or meta-mathematics is a scientific reflection and Knowledge about mathematics seen as an entity/ object in Human In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Abraham Robinson ( October 6, 1918 &ndash April 11, 1974) was a Mathematician who is most widely known for development of Non-standard Alfred Tarski ( January 14, 1901, Warsaw, Russian ruled Poland – October 26, 1983, Berkeley California Andrzej Mostowski ( 1 November 1913 – 22 August 1975) was a Polish Mathematician.

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 Maltsev in the 1940s went unnoticed because of the war. Anatoly Ivanovich Maltsev (Malcev ( Russian: Анато́лий Ива́нович Ма́льцев 27 November N 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. The International Congress of Mathematicians (ICM is the largest congress in the Mathematics community C. Chang, Leon Henkin, Bjarni Jónsson, R. Leon Henkin ( 19 April 1921 – 1 November[[ 006]] was a Logician at the University of California Berkeley. Bjarni Jónsson (born 1920 is an Icelandic Mathematician and Logician working in Universal algebra and Lattice theory. C. Lyndon, and others.

In the late 1950s, E. Marczewski^[3] 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 J. Mycielski, W. Narkiewicz, W. Nitka, J. Płonka, S. Świerczkowski, K. Urbanik, and others.

See also

Footnotes

References

• Bergman, George M. , 1998. An Invitation to General Algebra and Universal Constructions (pub. Henry Helson, 15 the Crescent, Berkeley CA, 94708) 398 pp. ISBN 0-9655211-4-1.
• Brainerd, Barron, Aug-Sep 1967. Review of Universal Algebra by P. M. Cohn. American Mathematical Monthly, 74(7): 878-880. The American Mathematical Monthly ( is a mathematical journal founded by Benjamin Finkel in 1894.
• Burris, Stanley N. , and H. P. Sankappanavar, 1981. A Course in Universal Algebra Springer-Verlag. ISBN 3-540-90578-2 Free online edition.
• Cohn, Paul Moritz, 1981. Universal Algebra. Dordrecht , Netherlands: D. Reidel Publishing. ISBN 90-277-1213-1 (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. ISBN 0-521-34832-3. Free online second edition.
• Grätzer, George, 1968. Universal Algebra D. Van Nostrand Company, Inc.
• Hobby, David, and Ralph McKenzie, 1988. The Structure of Finite Algebras American Mathematical Society. ISBN 0-8218-3400-2. Free online edition.
• Jipsen, Peter, and Henry Rose, 1992. Varieties of Lattices, Lecture Notes in Mathematics 1533. Springer Verlag. ISBN 0-387-56314-8. Free online edition.
• Pigozzi, Don. General Theory of Algebras.
• Smith, J. D. H. , 1976. Mal'cev Varieties, Springer-Verlag.
• Whitehead, Alfred North, 1898. Alfred North Whitehead, OM ( February 15 1861, Ramsgate, Kent, England &ndash December 30 1947, A Treatise on Universal Algebra, Cambridge. (Mainly of historical interest. )

External links

• Utility of Universal Algebra—comments on MacFarlane's 1899 review of Whitehead's Universal Algebra.

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