A formal system for manipulating terms over a signature by means of rules. A set R of rules (each a rewrite rule) creates an abstract reduction system →_R on the algebra T(Σ‎,X) of all terms over signature Σ‎ and variables X. Usually, the rules are a set E of equations that determine a reduction system →_E using rewrites based on equational logic.

Let E be a set of equations such that, for each t=t′ ∈‎ E, the left-hand side t is not a variable. The pair (Σ‎,E) is called an equational TRS. The equations of E are used in derivations of terms where the reduction t →_E t′ requires substitutions to be made in some equation e ∈‎ E and the left-hand side of e is replaced by the right-hand side of e in t to obtain t′.

The first set of properties of a term rewriting system (Σ‎,E) is now obtained from the properties of abstract reduction systems. The following are examples.

1. (1) The term rewriting system (Σ‎,E) is complete if the reduction system →_R on T(Σ‎) is Church-Rosser and strongly terminating.

2. (2) Let ≡_E be the smallest congruence containing →_E and T(Σ‎,E) be the factor algebra T(Σ‎)|≡_E. Then T(Σ‎,E) is the initial algebra of Alg(Σ‎,E). If (Σ‎,E) is a finite equational TRS specification that is complete, then T(Σ‎,E) is a computable algebra.