Universal algebra

Universal algebra is the field of mathematics that studies the ideas common to all algebraic systems[?].

Table of contents 1 Basic idea 2 Examples 2.1 Groups 2.2 Modules 3 Further issues

Basic idea

From the point of view of universal algebra, an algebra 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) is simply 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).

After the operations have been specified, the nature of the algebra can be further limited by axioms, which in universal algebra must 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.

According to Yde Venema[?], "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), and in which the language we use 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

Groups

To see how this is supposed to work, 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 group is problematic from the point of view of universal algebra. The reason is that the axioms of identity and inverse are not stated purely in terms of equational laws but also have clauses involving the phrase "there exists ... such that". This is not allowed in universal algebra. The solution is not difficult; we add a nullary operation e and a unary operation ~, in addition to the binary operation *, then list the axioms 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.)

Now, 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 new-fangled universal algebraists' groups might require more information than specifying one of the old-fashioned kind of groups. After all, nothing in the definition of group said that the identity element e was unique; if there is another identity element e', then it's ambiguous as to which should be the value of the nullary operator e. However, this is not a problem, because identity elements are always unique. The same thing is true of inverse elements. So the universal algebraist's definition of group really is equivalent to the usual definition.

Modules

later -- see Talk:Abstract algebra for a rough draft

Further issues

Once you have defined the operations and axioms for your algebra, you can now define the notion of homomorphism between two algebras A and B. A homomorphism h: A → B is simply a function 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 I have placed subscripts on f to indicate whether it's 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.

This article is too brief to indicate the breadth of the results of universal algebra. The motivation for the field is the many examples of algebras (in the sense of universal algebra), such as 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, you can prove them once and for all for every kind of algebraic system.

A more generalised programme along these lines is carried out by category theory. 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 just don't generalise all the way to category theory. Thus both fields of study are useful. The connection is that given a list of operations and axioms, the corresponding algebras and homomorphisms are the objects and morphisms of a category.