Editing Free object (section)

==Free universal algebras==
{{main|Term algebra}}
{{Expand section|date=June 2008}}

Let <math>S</math> be a set and <math>A</math> be an algebraic structure of type <math>\rho</math> generated by <math>S</math>. The underlying set of this algebraic structure <math>A</math>, often called its universe, is denoted by <math>A</math> . Let <math>\psi: S \to A</math> be a function. We say that <math>(A, \psi)</math> (or informally just <math>A</math>) is a free algebra of type <math>\rho</math> on the set <math>S</math> of free generators if the following universal property holds:

For every algebra <math>B</math> of type <math>\rho</math> and every function <math>\tau: S \to B</math>, where <math>B</math> is the universe of <math>B</math>, there exists a unique [[homomorphism]] <math>\sigma: A \to B</math> such that the following diagram commutes:

<math> \begin{array}{ccc} S & \xrightarrow{\psi} & A \\ & \searrow_{\tau} & \downarrow^{\sigma} \\ & & B \ \end{array} </math>

This means that <math>\sigma \circ \psi = \tau</math>.