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Let $A\neq \emptyset$ and $I$ be sets. An operation on the set $A$ is any function $f:A^{I}\rightarrow A$.

Can someone give non-trivial examples of operation. By trivial I mean the most given in books such as sum, subtraction, set inclusion, function composition, boolean sum, empty function, etc.


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I don't think what you mean by "non-trivial" is clear. For example, if we take set A to be $\{0, 1\}$ and I be a set of propositions, then we can think of operations as propositional logic formulas. Is that "non-trivial"? –  aelguindy Jan 17 '12 at 19:12
@aelguindy I think the OP is asking about examples of operators that he hasn't already seen in other contexts. –  Alex Becker Jan 17 '12 at 19:16
I have flagged this for moderator attention, as I think it should be community wiki. –  Gerry Myerson Jan 18 '12 at 4:40
I forgot that community wiki doesn't mean the same thing here that it means on MathOverflow. In fact, having reread a discussion of CW on the meta site, I no longer have any clear idea of what CW means here, and I wish to declare my neutrality as to whether this question should be CW. My apologies to all for wasting your time. –  Gerry Myerson Jan 18 '12 at 5:34

5 Answers 5

up vote 1 down vote accepted

Two examples of operations are infinite conjunction and disjunction on set of propositions: to be correct this should be regarded more like a family of operations.

Let $\{0,1\}$ be the set truth values, for each cardinal $\aleph$ you can consider the operation $$\bigwedge \colon \mathcal \{0,1\}^\aleph \to \mathcal \{0,1\}$$ this operation is such that for every family of truth values $(x_i)_{i \in \aleph} \in \{0,1\}^\aleph$, $\bigwedge_{i \in \aleph} x_i = 1$ if and only if $x_i=1$ for each $i \in \aleph$.

In similar way you can consider the operation $$\bigvee \colon \{0,1\}^\aleph \to \{0,1\}$$ such that for every family $(x_i)_{i \in \aleph} \in \{0,1\}^\aleph$ the equality $\bigvee_{i \in \aleph} x_i = 1$ if $x_i=1$ for at least one $i \in \aleph$.

Another operation which is important (at least in my opinion) is the juxtaposition of words. Consider an arbitrary set $\Sigma$ then you can define over the set $\Sigma^*=\bigcup_{n \in \mathbb N} \Sigma^n$ the operation $$\cdot\colon {\Sigma^*}^2 \to \Sigma$$ defined as $$\cdot \left((x_i)_{i=1,\dots,n},(y_i)_{i=1,\dots,m}\right) = (z_i)_{i=1,\dots,n+m}$$ where $$z_i = x_i$$ if $i \leq n$ and $$z_i = y_{i-n}$$ otherwise. As I said this is a pretty important operation because the set $\Sigma^*$ with this operation gives us an example of free monoid, which is in a certain sense a prototypical monoid, in which we are able to explicit write computations.

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Here's a fun (and useful!) example when $I$ is infinite. Let $A$ be a compact Hausdorff space. For any ultrafilter on $I$, we can define an operation $A^I \to A$ called the limit which generalizes the limit of a sequence. Unlike the limit of a sequence, the limit in this context is guaranteed to exist (by compactness) and be unique (by Hausdorffness), and in fact these conditions are reversible: a compact Hausdorff space is precisely an object for which it is possible to assign limits in this way consistently. (More precisely, it's precisely an algebra over a monad called the ultrafilter monad.)

In other words, ultrafilters allow us to think about compact Hausdorff spaces as "algebraic" objects in a certain sense, where all the interesting algebraic operations are infinitary! (The only ultrafilters on finite sets are the principal ones, and the corresponding limits are just the projections.)

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"Trivial" is not a precise word, but perhaps you'll agree that this example is non-trivial. Let $A=\mathbb{N}$, the natural numbers, and let $I$ be a three-element set. Note that $A^I\cong A\times A\times A$. Define the operation $f:A\times A\times A\to A$ by $$f(x,y,z)=\#\text{ of ways of expressing }x\text{ as a sum of }y\text{ }z^{\text{th}}\text{ powers}.$$ See here for some info about $f(x,y,2)$.

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One interesting infinitary operation that hasn’t been mentioned is the Suslin operation, sometimes (following Suslin) called operation A, which is important in descriptive set theory. If $\mathscr{F}=\{F_\sigma:\sigma\in^{<\omega}\omega\}$ is a family of sets indexed by the set of finite sequences of natural numbers, the result of applying the Suslin operation to $\mathscr{F}$ is $$\bigcup_{\sigma\in^\omega\omega}\;\;\bigcap_{n\in\omega}\;F_{\sigma\upharpoonright n}\;.$$

Added: I should probably note that in terms of the notation in the question, $I$ here is $^{<\omega}\omega$, and $A$ is some family of sets, e.g., the family of closed sets in a metric space.

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«Operation A» is a terrible name! –  Mariano Suárez-Alvarez Jan 18 '12 at 4:22
@Mariano: I agree, but the name hasn’t wholly disappeared. According to Aki Kanamori, Suslin named the operation after Alexandrov. –  Brian M. Scott Jan 18 '12 at 4:35
@Mariano: Would you prefer "Operation Desert Storm"? –  Asaf Karagila Jan 18 '12 at 10:38

If you want an example with infinite $I$, you can let $A=[0,1]$ and $I=\mathbb N$. Then given a map $h:\mathbb N\rightarrow [0,1]$, we can define $f(h)=\limsup_n h(n)$.

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