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. 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. 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 ⋀ α ∈ J x α {displaystyle extstyle igwedge _{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 Ω {displaystyle Omega } , where Ω {displaystyle Omega } 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 is further defined 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. A collection of algebraic structures defined by identities is called a variety or equational class. Some authors consider varieties to be the main focus of universal algebra.