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I looked at here, and here, and here, to understand the terminology used regarding indexed family of sets (but I still left with some questions):

Take S as the set $S=\{1,2,3\}$

we define:

1) K to be an S-indexed family of operation symbols.

2) K to be a family of operation symbols indexed by S.

My Questions:

1) Do the above two statements mean the same?

2) If the answer to the question one is "yes", do both mean $K=\{F_1,F_2,F_3\}$, where each $F_i$ is a set of operation symbols, or do they mean $K=\{f_1,f_2,f_3\}$, where each $f_i$ is just an operation symbol?

3) If they mean $K=\{F_1,F_2,F_3\}$ (in the question two), couldn't this be clarified by saying "K is an S-indexed family of sets of operation symbols."?

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qartal
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1 Answers1

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Yes, the two wordings mean the same. In both case $K$ has the shape $(f_1,f_2,f_3)$, where each of the $f_i$s is an operation symbol.

Formally, this just means that $K$ is some map* with domain $S$ such that each value of the map is an operation symbol. Writing them as, say, $f_2$ instead of $K(2)$ is just a notational choice that may make formulas that involve the symbols easier to read.

In many cases where one says something like "an $S$-indexed family of operation symbols", it is implied that different elements of $S$ correspond to different operation symbols (in other words, that the map is injective), but this is not strictly required by the wording "indexed family" and in practice it is up to the reader to figure out whether such an assumption makes better sense in context than allowing some of the indexed symbols to be identical.

In particular, the assumption that the elements of the family are different for different indices will often be in force when it's a family of symbols in logic or formal language theory.

*: In some contexts the wording "$S$-indexed family" may be used where $S$ is a proper class rather than a set. Then the family itself must be a proper class, which one may or may not consider to qualify as a "map".

hmakholm left over Monica
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  • It is strange. The context I ask this question is Algebra. I am reading this, page 6: it says that $\Sigma={ \Sigma_{w,k}}{(w,k)\in K^\times K}$ is a ($K^ \times K$)-indexed family of function symbols. The writer definitely mean that elements of $\Sigma$ are sets of function symbols, not function symbols (i.e., $\Sigma{w,k}$ are sets of symbols not just a symbol), since in the footnote of the same page indicate this. – qartal May 16 '15 at 16:16
  • Actually I see in another resource, (see page 3) the writer strangely mean "a family of sets of function symbols, by writing "A family of functions indexed by ($K^* \times K$). I am wondering if this is norm, or I am missing something! – qartal May 16 '15 at 16:27
  • @qartal: Individually I wouldn't hesitate to call each of your examples a typo; the author clearly intended to make $\Sigma$ a "$(K^*\times K)$-indexed family of sets of symbols" -- and I would say I'm used to reading definitions in the general area of these two papers. However, since you're found two examples it leads me to wonder if there's a subcommunity that uses "indexed family" in a different sense that what I'm used to. – hmakholm left over Monica May 16 '15 at 16:38