Universal Algebra

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See: General Algebra, Order Theory.



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.

    • From the point of view of universal algebra, an algebra (or algebraic structure) is a set [math]\displaystyle{ A }[/math] together with a collection of operations on A. An n-ary operation on [math]\displaystyle{ A }[/math] is a function that takes [math]\displaystyle{ n }[/math] elements of [math]\displaystyle{ A }[/math] and returns a single element of A. Thus, a 0-ary operation (or nullary operation) can be represented simply as an element of [math]\displaystyle{ A }[/math], or a constant, often denoted by a letter like a. A 1-ary operation (or unary operation) is simply a function from [math]\displaystyle{ A }[/math] to [math]\displaystyle{ A }[/math], 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 [math]\displaystyle{ f }[/math](x,y,z) or [math]\displaystyle{ f }[/math](x1,...,xn). Some researchers allow infinitary operations, such as [math]\displaystyle{ \textstyle\bigwedge_{\alpha\in J} x_\alpha }[/math] 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 [math]\displaystyle{ \Omega }[/math], where [math]\displaystyle{ \Omega }[/math] is an ordered sequence of natural numbers representing the arity of the operations of the algebra.