Universal algebra

Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes "the theory of groups" as an object of study.

Basic idea

In 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 $\textstyle \bigwedge _{\alpha \in J}x_{\alpha }$ 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 $\Omega$, where $\Omega$ 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 can be further limited by axioms, which in universal algebra often take the form of identities, or 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.

Varieties

An algebraic structure that can be defined by identities is called a variety, and these are sufficiently important that some authors consider varieties the only object of study in universal algebra, while others consider them an object.

Restricting one's study to varieties rules out:

• quantification, including universal quantification ($\forall$), except before an equation, and existential quantification ( $\exists$ )
  • Predicate logic in particular is ruled out
• All relations except equality, in particular inequalities, both a ≠ b and order relations

In this narrower definition, universal algebra 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.

Not all algebraic structures in a wider sense fall into this scope. For example, ordered groups are not studied in mainstream universal algebra because they involve an ordering relation.

A more fundamental restriction is that universal algebra cannot study the class of fields, because there is no type (a.k.a. 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 simply be added to the type).

One advantage of this restriction is that the structures studied in universal algebra can be defined in any category that has finite products. For example, a topological group is just a group in the category of topological spaces.

Examples

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

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 section): x ∗ (y ∗ z)  =  (x ∗ y) ∗ z;   formally: ∀x,y,z. x∗(y∗z)=(x∗y)∗z.
• Identity element: There exists an element e such that for each element x, e ∗ x  =  x  =  x ∗ e;   formally: ∃e ∀x. e∗x=x=x∗e.
• Inverse element: It can easily be seen that the identity element is unique. If this unique identity element is 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 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 by calling ∗ a binary operation.)

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 by adding 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; formally: ∀x. e∗x=x=x∗e.
• Inverse element: x ∗ (~x)  =  e  =  (~x) ∗ x   formally: ∀x. 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.)

What has changed is that in the usual definition there are:

• a single binary operation (signature (2))
• 1 equational law (associativity)
• 2 quantified laws (identity and inverse)

...while in the universal algebra definition there are

• 3 operations: one binary, one unary, and one nullary (signature (2,1,0))
• 3 equational laws (associativity, identity, and inverse)
• no quantified laws (except for outermost universal quantifiers which are allowed in varieties)

It is 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 is ambiguous which one should be the value of the nullary operator e. Proving that it is unique is a common beginning exercise in classical group theory textbooks. The same thing is true of inverse elements. So, the universal algebraist's definition of a group is equivalent to the usual definition.

At first glance this is simply a technical difference, replacing quantified laws with equational laws. However, it has immediate practical consequences – 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), and cannot use quantified laws (which do not make sense, as objects in general categories do not have elements). Further, the perspective of universal algebra insists not only that the inverse and identity exist, but that they be maps in the category. The basic example is of a topological group – not only must the inverse exist element-wise, but the inverse map must be continuous (some authors also require the identity map to be a closed inclusion, hence cofibration, again referring to properties of the map).

Basic constructions

We assume that the type, $\Omega$, 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_A of A and corresponding f_B of B (of arity, say, n), h(f_A(x₁,...,x_n)) = f_B(h(x₁),...,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 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 cartesian product of the sets with the operations defined coordinatewise.

Some basic theorems

• The isomorphism theorems, which encompass 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 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 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.

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.

Generalizations

A more generalised programme 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, many theorems that hold in universal algebra do not generalise all the way to category theory. Thus both fields of study are useful.

A more recent development in category theory that generalizes operations is operad theory – an operad is a set of operations, similar to a universal algebra.

Another development is partial algebra where the operators can be partial functions.

An important generalization of universal algebra theory is model theory, which is sometimes described as "universal algebra + logic".

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.^[1]

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 Ausdehnungslehre, 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."^[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. 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, Roger Lyndon, and others.

In the late 1950s, Edward 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 Jan Mycielski, Władysław Narkiewicz, Witold Nitka, J. Płonka, S. Świerczkowski, K. Urbanik, and others.

See also

• Graph algebra
• Homomorphism
• Lattice theory
• Signature
• Term algebra
• Variety
• Clone
• Universal algebraic geometry
• Model theory

Footnotes

1. Grätzer, George. Universal Algebra, Van Nostrand Co., Inc., 1968, p. v.
2. Quoted in Grätzer, George. Universal Algebra, Van Nostrand Co., Inc., 1968.
3. Marczewski, E. "A general scheme of the notions of independence in mathematics." Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 6 (1958), 731–736.

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.
• Birkhoff, Garrett, 1946. Universal algebra. Comptes Rendus du Premier Congrès Canadien de Mathématiques, University of Toronto Press, Toronto, pp. 310–326.
• Brainerd, Barron, Aug–Sep 1967. Review of Universal Algebra by P. M. Cohn. American Mathematical Monthly, 74(7): 878–880.
• 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.
• 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. 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. Free online edition.
• Smith, J.D.H., 1976. Mal'cev Varieties, Springer-Verlag.
• Whitehead, Alfred North, 1898. A Treatise on Universal Algebra, Cambridge. (Mainly of historical interest.)

External links

• Algebra Universalis—a journal dedicated to Universal Algebra.