Then , together with the set of constants and -ary operations defined above is an -structure (http://planetmath.org/Structure). Since there are no relations defined on it, is an algebraic system whose signature is the subset of consisting of all but the relation symbols of . The algebra is aptly called the term algebra of the signature (over ).
The prototypical example of a term algebra is the set of all well-formed formulas over a set of propositional variables in classical propositional logic. The signature is just the set of logical connectives. For each -ary logical connective , there is an associated -ary operation on , given by .
Remark. The term algebra of a signature over a set of variables can be thought of as a free structure in the following sense: if is any -structure, then any function can be extended to a unique structure homomorphism . In this regard, can be viewed as a free basis for the algebra . As such, is also called the absolutely free -structure with basis .
|Date of creation||2013-03-22 17:35:24|
|Last modified on||2013-03-22 17:35:24|
|Last modified by||CWoo (3771)|