Troubles by understanding arity of terms in A Course in Universal Algebra.

Question

In A Course in Universal Algebra in definition 10.1 terms are introduced.

What puzzles me is the statement about arity:

"A term $p$ is $n$-ary if the number of variables appearing explicitly in $p$ is $\leq n$."

Does that mean that the arity of a term $p$ is at most limited but not defined, and can take several values?

If e.g. $p$ is a variable then can every integer $\geq1$ be put forward as arity of $p$?

This seems to be confirmed in the definition 10.2 (also troubling me) where for term $p(x_1,\dots,x_n):=x_i\in X$ a mapping $p^{\mathbf A}:A^n\to A$ is prescribed by $(a_1,\dots,a_n)\mapsto a_i$.

I really do not understand this definition.

If I just start with term $x$ wich is a variable then according to def. 10.2 $x^{\mathbf A}$ must be some function $A^n\to A$. But if so, then what is $n$ (can it take several values?) and how is the function prescribed?

Answer 1

I think you've got it (ugly as it seems). The point is that a term on its own isn't a "complete" object, and so it doesn't have a unique arity.

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Certainly there's no problem talking about the arity of a map from $A^n$ to $A$ (namely, just $n$). The point is that a single term can have multiple different interpretations as functions, and these yield different arities. You point this out yourself.

Specifically, for any term $p$ and any set $V$ of variables such that every variable occurring in $p$ is in $V$, we can view $p$ as a map $p_V: A^V\rightarrow A$. Note that the translation $(p,V)\mapsto p_V$ is completely straightforward; this is why one will (sadly) often conflate $p$ and $p_V$ when $V$ is clear from context.

• Note that there's no requirement here that $V$ be finite!

In light of this, it isn't too weird to allow a term to have multiple arities - the point being that in some sense a term on its own is incomplete (for spiritual similarity, one might argue that the graph of a function is incomplete since it doesn't tell you the codomain).

Answer 2

Personally, I wouldn't even define arity for a term; I would only define it for a term operation. Let me try to be more precise:

Let $X$ be a set of symbols, called variables, and let $\mathcal{F}$ be a set of operation symbols such that each operation symbol $f$ has an arity, $\mathrm{ar}(f)$, that is a non-negative integer. The set $\mathrm{T}(X)$ of all terms in $X$ is defined recursively:

1. every variable $x$ or operation symbol $c$ with $\mathrm{ar}(c)=0$ is in $\mathrm{T}(X)$,
2. if $t_1,\dots,t_k$ are terms and $f$ is an operation symbol with $\mathrm{ar}(f)=n$, then $f(t_1,\dots,t_n)$ is in $\mathrm{T}(X)$.

Now, let $X_k=\{x_1,\dots,x_k\}$ and let $\mathbf{A}$ be an algebra of type $\mathcal{F}$. To every term $t$ in $\mathrm{T}(X_k)$ we can define a $k$-ary operation $t^A$ on $A$ as follows:

1. if $t=x_i$ or $t=c$ with $\mathrm{ar}(c)=0$, then $t^A:A^k\to A$ is defined by $t^A(a_1,\dots,a_k)=a_i$ or $t^A(a_1,\dots,a_k)=c^A$, respectively.
2. if $t=f(t_1,\dots,t_n)$, then $t^A(a_1,\dots,a_k)=f^A(t_1^A(a_1,\dots,a_k),\dots,t_n^A(a_1,\dots,a_k))$.

The important thing to notice is that terms are just formal strings of symbols whereas when we assign meaning to a term in a specific algebra that is how we arrive at term operation. Based on this, I feel no reason to define arity for a term and the arity of a term operation is just the arity of the operation. I will conclude with an example:

Let $\mathcal{F}=\{+,-,0\}$. Then $t=(x_1+(-x_2))+x_3$ is in $\mathrm{T}(X_3)$, but it is also in $\mathrm{T}(X_k)$ for any $k\geq 3$. So, given an algebra $\mathbf{A}$ of type $\mathcal{F}$, we can define the term operation $t^A:A^3\to A$ by $t(a_1,a_2,a_3)=(a_1+(-a_2))+a_3$ or we could define it $t^A:A^k\to A$ for any $k\geq 3$ by $t(a_1,\dots,a_k)=(a_1+(-a_2))+a_3$. This first term operation has arity equal to 3 and the second one has arity equal to $k$, but there is no reason to define arity for $t$.