Foundation of the Rewriting in an Algebra

This paper includes two main ideas. The first one, rewriting in an algebra, was introduced in [5]. The second one, boolean rewriting, can be found in many papers but we were never able to find a clear comparison with the classic one. We prefer rewriting in an algebra to term rewriting. This is our way to give a unique theory of rewriting. If the algebra is free, then we get the term rewriting. If the algebra is a certaine quotient of a free algebra then we get rewriting modulo equations. Rewriting is said to be boolean when the condition of each conditional equation is of boolean sort(in the free algebra it is a boolean term). We prove the classic rewriting is equivalent to boolean rewriting in a specific algebra, therefore, boolean rewriting is more general than the classic one.