Natural examples of reflexive, symmetric, transitive closure of a relation

Question

What are some simple yet "natural" examples of specifying an equivalence relation by giving just the "essential" or "basic" identifications while leaving implicit all the others that entails. In other words, "natural" examples of a relation $\alpha$ whose reflexive, symmetric, transitive closure is an equivalence relation.

One such example arises in elementary topology where one constructs the line-with-two-origins by saying something like the following: "Define an equivalence relation $\sim$ on the set $\mathbb{R} \times {0, 1}$ by taking $(x, 0) \sim (x, 1)$ for each $x \neq 0$."

I'm looking for really simple examples other than that and the obvious ones used to obtain a cylinder, cone, double cone, Möbius strip, torus, and projective plane from the unit square.

(Of course it is trivial to start with a small finite set to manufacture artificial examples.)

Answer 1

An example is modular arithmetic. Assert $x \sim x+6$ for all $x \in \Bbb{Z}$ and you obtain $\Bbb{Z}/6\Bbb{Z}$ as the reflexive, symmetric, and transitive closure. ($6$ is not essential. One can get any congruence relation on the integers by varying the $6$.)

Answer 2

Here's another fun one, defined by Witt for quadratic forms over a field $K$. Let me assume the characteristic of $K$ is not 2.

Two $n$-tuples of nonzero elements $(a_1, \ldots, a_n), (b_1, \ldots, b_n) \in (K -\{0\})^n$ are said to be simply chain equivalent if there are indices $i, j$ such that

1. the quadratic forms $\langle a_i, a_j \rangle$ $\langle b_i, b_j \rangle$ are isometric, and
2. for $k \not \in \{i,j\}$, we have $a_k = b_k$.

This is not an equivalence relation. The equivalence relation on $(K -\{0\})^n$ generated by this relation is called chain equivalence.

The cool thing, proved by Witt, is that $(a_1, \ldots, a_n)$ and $(b_1, \ldots, b_n)$ are chain equivalent if and only if $\langle a_1, \ldots, a_n \rangle$ and $\langle b_1, \ldots, b_n \rangle$ are isometric.

Answer 3

Define the relation $\sim$ on $\mathbb{Z} \times (\mathbb{Z} \setminus \{ 0 \})$ by $\{ (a, b), (ac, bc) \mid a \in \mathbb{Z}; b, c \in \mathbb{Z} \setminus \{ 0 \} \}$. Then the generated equivalence relation turns out to be exactly the same as $\{ ((a, b), (c, d)) \mid ad = bc \}$ which is commonly used in defining the rational numbers $\mathbb{Q}$ as the quotient by this equivalence relation. In other words, the rational numbers can be described as the formal fractions $\frac{a}{b}$ subject to the requirement that you want $\frac{a}{b}$ and $\frac{ac}{bc}$ to be equivalent fractions in the end -- and no further requirements need to be imposed.

In commutative algebra, a similar construction holds more generally for localization rings $S^{-1} R$ and localization modules $S^{-1} M$. (And in fact, it can be useful in describing why in the general case where $S$ can contain zero divisors, the more general definition of localization needs to become a bit more fiddly.)