universal algebra

 

Universal algebra (sometimes called General algebra) is the field of mathematics that studies the ideas common to all algebraic structures.

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. 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. A 2-ary operation (or binary operation) is often denoted by a symbol placed between its arguments, like x * y. 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. 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. An example is the associative 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.

Universal algebra can be seen as a special branch of model theory, in which we are dealing with structures having operations only (i.e., no relations except for equality), and in which the language used to talk about these structures uses equations only. On the other hand the structures are such that they can be defined in any category which has finite products.

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. Normally a group is defined in terms of a single binary operation *, subject to these axioms:

• Associativity (as in the previous paragraph): x * (y * z)  =  (x * y) * z.
• Identity element: There exists an element e such that e * x  =  x  =  x * e.
• Inverse element: For each x, there exists an element i such that x * i  =  e  =  i * x.

(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.)

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. The same thing is true of inverse elements. 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,...,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 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 cross product of the sets with the operations defined coordinatewise.

Some basic theorems

• The Isomorphism theorems, which encompasses the Isomorphism theorems of groups, rings, modules, etc.
• 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.

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. 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 his 1963 thesis^[1], William Lawvere showed that every algebraic theory corresponds to a cartesian category (i.e. given objects X and Y, there is a product object X × Y), and conversely, every cartesian category can be expressed in terms of an algebraic theory. A functor from the category into the category of sets assigns a set to each type and a function to each function symbol satisfying the axioms. Every group, for instance, arises as a functor F:Th(Grp)→ Set, where Th(Grp), is the theory of groups described above.

Further issues

A more generalised program along these lines is carried out by category theory. Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a category. Category theory applies to many situations where universal algebra does not, extending the reach of the theorems. Conversely, some 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. Whitehead credits William Rowan Hamilton and Augustus De Morgan as originators of the subject matter, and James Joseph Sylvester with coining the term itself^[2].

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 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." 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.

Whitehead's early work sought to unify quaternions (due to Hamilton), Grassmann's calculus of extensions, 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."^[3]

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. 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 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. 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, R. C. Lyndon, and others.

In the late 1950s, E. Marczewski^[4] 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

• category theory
• homomorphism
• lattice theory
• signature
• variety
• universal
• operad theory

Footnotes

References

• Bergman, George M.: An Invitation to General Algebra and Universal Constructions (pub. Henry Helson, 15 the Crescent, Berkeley CA, 94708) 1998, 398 pp. ISBN 0-9655211-4-1.
• Brainerd, Barron, review of Universal Algebra by P. M. Cohn in The American Mathematical Monthly, Vol. 74, No. 7. (Aug. - Sep., 1967), pp. 878-880.
• Burris, Stanley N., and H.P. Sankappanavar: A Course in Universal Algebra Springer-Verlag, 1981 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: Universal Algebra D. Van Nostrand Company, Inc., 1968.
• Hobby, David, and Ralph McKenzie, 1988. The Structure of Finite Algebras American Mathematical Society. ISBN 0-8218-3400-2. (Available free online as of May 31, 2006).
• Jipsen, Peter, and Henry Rose, 1992. Varieties of Lattices, Lecture Notes in Mathematics 1533. Springer Verlag. ISBN 0-387-56314-8. Free online edition.
• Smith, J.D.H. Mal'cev Varieties, Springer-Verlag, 1976.

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