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term algebra (Definition)

Let $ \Sigma$ be a signature and $ V$ a set of variables. Consider the set of all terms of $ T:=T(\Sigma)$ over $ V$. Define the following:

  • For each constant symbol $ c\in \Sigma$, $ c^T$ is the element $ c$ in $ T$.
  • For each $ n$ and each $ n$-ary function symbol $ f\in \Sigma$, $ f^T$ is an $ n$-ary operation on $ T$ given by
    $\displaystyle f^T(t_1,\ldots,t_n)=f(t_1,\ldots,t_n),$
    meaning that the evaluation of $ f^T$ at $ (t_1,\ldots,t_n)$ is the term $ f(t_1,\ldots, t_n)\in T$.
  • For each relational symbol $ R\in \Sigma$, $ R^T=\varnothing$.

Then $ T$, together with the set of constants and $ n$-ary operations defined above is an $ \Sigma$-structure. Since there are no relations defined on it, $ T$ is an algebraic system whose signature $ \Sigma'$ is the subset of $ \Sigma$ consisting of all but the relation symbols of $ \Sigma$. The algebra $ T$ is aptly called the term algebra of the signature $ \Sigma$ (over $ V$).

Remark. The term algebra $ T$ of a signature $ \Sigma$ over a set $ V$ of variables can be thought of as a free structure in the following sense: if $ A$ is any $ \Sigma$-structure, then any function $ \phi:V\to A$ can be extended to a unique structure homomorphism $ \phi':T\to A$. In this regard, $ V$ can be viewed as a free basis for the algebra $ T$. As such, $ T$ is also called the absolutely free $ \Sigma$-structure with basis $ V$.



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See Also: polynomials in algebraic systems, free algebra

Other names:  word algebra
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Cross-references: basis, free basis, structure homomorphism, function, algebra, relation symbols, subset, algebraic system, relations, operation, function symbol, constant symbol, terms, variables, signature
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This is version 4 of term algebra, born on 2007-10-16, modified 2007-12-11.
Object id is 10002, canonical name is TermAlgebra.
Accessed 757 times total.

Classification:
AMS MSC: 03C99 (Mathematical logic and foundations :: Model theory :: Miscellaneous)
 03C60 (Mathematical logic and foundations :: Model theory :: Model-theoretic algebra)
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