# Universal Algebra

• From the point of view of 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 $\textstyle\bigwedge_{\alpha\in J} x_\alpha$ 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.