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

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For each constant symbol $c\in \mathrm{\Sigma}$, ${c}^{T}$ is the element $c$ in $T$.

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For each $n$ and each $n$ary function symbol $f\in \mathrm{\Sigma}$, ${f}^{T}$ is an $n$ary operation^{} on $T$ given by
$${f}^{T}({t}_{1},\mathrm{\dots},{t}_{n})=f({t}_{1},\mathrm{\dots},{t}_{n}),$$ meaning that the evaluation of ${f}^{T}$ at $({t}_{1},\mathrm{\dots},{t}_{n})$ is the term $f({t}_{1},\mathrm{\dots},{t}_{n})\in T$.

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For each relational symbol $R\in \mathrm{\Sigma}$, ${R}^{T}=\mathrm{\varnothing}$.
Then $T$, together with the set of constants and $n$ary operations defined above is an $\mathrm{\Sigma}$structure^{} (http://planetmath.org/Structure). Since there are no relations^{} defined on it, $T$ is an algebraic system whose signature ${\mathrm{\Sigma}}^{\prime}$ is the subset of $\mathrm{\Sigma}$ consisting of all but the relation symbols of $\mathrm{\Sigma}$. The algebra^{} $T$ is aptly called the term algebra of the signature $\mathrm{\Sigma}$ (over $V$).
The prototypical example of a term algebra is the set of all wellformed formulas over a set $V$ of propositional variables in classical propositional logic^{}. The signature $\mathrm{\Sigma}$ is just the set of logical connectives. For each $n$ary logical connective $\mathrm{\#}$, there is an associated $n$ary operation $[\mathrm{\#}]$ on $V$, given by $[\mathrm{\#}]({p}_{1},\mathrm{\dots},{p}_{n})=\mathrm{\#}{p}_{1}\mathrm{\cdots}{p}_{n}$.
Remark. The term algebra $T$ of a signature $\mathrm{\Sigma}$ over a set $V$ of variables can be thought of as a free structure in the following sense: if $A$ is any $\mathrm{\Sigma}$structure, then any function $\varphi :V\to A$ can be extended to a unique structure homomorphism^{} ${\varphi}^{\prime}:T\to A$. In this regard, $V$ can be viewed as a free basis for the algebra $T$. As such, $T$ is also called the absolutely free $\mathrm{\Sigma}$structure with basis $V$.
Title  term algebra 

Canonical name  TermAlgebra 
Date of creation  20130322 17:35:24 
Last modified on  20130322 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 