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According to the definitions in Sankappanavar's universal algebra :

Assume $p$ is a term, then $p(x_1,x_2,...,x_n)$ indicates that the variables occurring in $p$ are among $x_1,...,x_n$. But there is an ambiguity : when $p=x_1$, for example, we can write it as $p(x_1)$, $p(x_1,x_2)$ and $p(x_2,x_1)$, and so on, but then they correspond to different term functions, say $p[x_1](a)=a$, $p[x_1,x_2](a,b)=a$ and $p[x_2,x_1](a,b)=b$, here I use $p[...]$ to indicate the corresponding $p(x_1,...,x_n)$ expression. So how to deal with such situation?

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  • $\begingroup$ Why do you think there is a need to "deal" with it? What exactly is the problem? First, when you discuss, say, the set of $n$-ary term functions, you fix a sequence of variables, say $(x_1,\dots,x_n)$ in advance, and consider term in these variables, hence there is no ambiguity. In any case, the sets of terms and term functions are closed under permutations of variables, hence the ambiguity does not matter. $\endgroup$
    – Emil Jeřábek
    Commented Oct 5, 2023 at 6:38
  • $\begingroup$ I don't know is $X$ allowed to be $\{x\}$, in this case, all terms are of the the form $p(x)$, and obviously the term functions can no longer 'generate' $Sg(X)$, when considering an algebra $A$ and its subset $X$. $\endgroup$
    – BAD MAN
    Commented Oct 5, 2023 at 6:53
  • $\begingroup$ And the question came to my mind when I read the Definition 11.1 in Chapter 2. It is about identities and things like $A\models p\approx q$, and there is an obvious ambiguity when the tuple $(x_1,...,x_n)$ is not clear. Also, when I went back to the proof of Theorem 10.3(c) in Chapter 2, I find it hard to show $E^{k}(S)$ is equal to RHS, because of similar problems. $\endgroup$
    – BAD MAN
    Commented Oct 5, 2023 at 7:02
  • $\begingroup$ I undertand now, I misunderstood the definition. $\endgroup$
    – BAD MAN
    Commented Oct 6, 2023 at 2:59
  • $\begingroup$ Good to hear that. $\endgroup$
    – Emil Jeřábek
    Commented Oct 6, 2023 at 7:09

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