term algebra
Let be a signature and a set of variables. Consider the set of all terms of over . Define the following:
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 .
Title | term algebra |
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Canonical name | TermAlgebra |
Date of creation | 2013-03-22 17:35:24 |
Last modified on | 2013-03-22 17:35:24 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 9 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 03C99 |
Classification | msc 03C60 |
Synonym | word algebra |
Related topic | PolynomialsInAlgebraicSystems |
Related topic | FreeAlgebra |