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Freely generated algebraic structure over a given signature
In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature. For example, in a signature consisting of a single binary operation, the term algebra over a set X of variables is exactly the free magma generated by X. Other synonyms for the notion include absolutely free algebra and anarchic algebra. From a category theory perspective, a term algebra is the initial object for the category of all X generated algebras of the same signature, and this object, unique up to isomorphism, is called an initial algebra; it generates by homomorphic projection all algebras in the category. A similar notion is that of a Herbrand universe in logic, usually used under this name in logic programming, which is (absolutely freely) defined starting from the set of constants and function symbols in a set of clauses. That is, the Herbrand universe consists of all ground terms: terms that have no variables in them. An atomic formula or atom is commonly defined as a predicate applied to a tuple of terms; a ground atom is then a predicate in which only ground terms appear. The Herbrand base is the set of all ground atoms that can be formed from predicate symbols in the original set of clauses and terms in its Herbrand universe. These two concepts are named after Jacques Herbrand. Term algebras also play a role in the semantics of abstract data types, where an abstract data type declaration provides the signature of a multi sorted algebraic structure and the term algebra is a concrete model of the abstract declaration.

A type is a set of function symbols, with each having an associated arity (i. e. number of inputs). For any non negative integer , let denote the function symbols in of arity A constant is a function symbol of arity 0. Let be a type, and let be a non empty set of symbols, representing the variable symbols. (For simplicity, assume and are disjoint.) Then the set of terms of type over is the set of all well formed strings that can be constructed using the variable symbols of and the constants and operations of Formally, is the smallest set such that:
  — each variable symbol from is a term in , and so is each constant symbol from For all and for all function symbols and terms , we have the string   — given terms , the application of an ary function symbol to them represents again a term. The term algebra of type over is, in summary, the algebra of type that maps each expression to its string representation. Formally, is defined as follows:
The domain of is For each nullary function in , is defined as the string For all and for each n ary function in and elements in the domain, is defined as the string
A term algebra is called absolutely free because for any algebra of type , and for any function , extends to a unique homomorphism , which simply evaluates each term to its corresponding value Formally, for each :
If , then If , then If where and , then .

The signature σ of a language is a triple <O, F, P> consisting of the alphabet of constants O, function symbols F, and predicates P. The Herbrand base of a signature σ consists of all ground atoms of σ: of all formulas of the form R(t1,  , tn), where t1,  , tn are terms containing no variables (i. e. elements of the Herbrand universe) and R is an n ary relation symbol (i. e. predicate). In the case of logic with equality, it also contains all equations of the form t1 = t2, where t1 and t2 contain no variables.

Term algebras can be shown decidable using quantifier elimination. The complexity of the decision problem is in NONELEMENTARY because binary constructors are injective and thus pairing functions.