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 class 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,...,xn). 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 , where is an ordered sequence of natural numbers representing the arity of the operations of the algebra.

It is also possible to define an algebra via the relations in the algebra instead of the operations. Birkhoff's Theorem states that the two definitions are equivalent, i.e., there is a Galois connection between relational and operational structures. This connection can be easily illustrated on the case of lattices, where the algebraic structure can be given by the operations join and meet or by introducing a partial order relation. The relational point of view is useful in computational problems, in particular for the constraint satisfaction problem (CSP).

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.