# Non-trivial examples of operations.

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.

Thanks.

• 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
• – dls Jan 17 '12 at 19:13
• @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

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.

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.)

"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 $f(x,y,2)$.

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.

• «Operation A» is a terrible name! – Mariano Suárez-Álvarez 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 $f(h)=\limsup_n h(n)$.