I thought I might bring together all of the examples of posets that people have proposed:

1. [#4 John Baez](https://forum.azimuthproject.org/discussion/comment/15887/#Comment_15887): the set real numbers \\(\mathbb{R}\\), with the usual notion of \\(\le\\)

2. [#5 Dan Schmidt](https://forum.azimuthproject.org/discussion/comment/15888/#Comment_15888): the set of pairs \\( (x,y) \in \mathbb R^2 \\) where \\( a \leq b \\) when \\( a_x \leq b_x \\) and \\( a_y \leq b_y \\).

3. [#15 Matthew Doty](https://forum.azimuthproject.org/discussion/comment/15951/#Comment_15951):

* Java classes form a *preordered set* when ordered by inheritance.

* Natural numbers \\(\mathbb{N}\\) form a *traditional poset* under the relation "\\(x\\) divides \\(y\\)", often denoted "\\(x\ |\ y\\)".

* For two sets \\(A,B \subseteq \mathbb{N} \\), we say \\(A\\) *is Turing reducible to* \\(B\\) if there is an [oracle machine](https://en.wikipedia.org/wiki/Oracle_machine) with oracle tape containing \\(B\\) that can compute the characteristic of \\(A\\). This is denoted \\(A \leq_T B\\). Under \\(\leq_T\\), \\(\mathcal{P}(\mathbb{N})\\) is a *preordered set*.

4. [#20 Alex Ortiz](https://forum.azimuthproject.org/discussion/comment/15965/#Comment_15965): An example of posets from analysis. If a measure space \\((X,\mu)\\) is \\(\sigma\\)-finite, then its \\(L^p\\) spaces (the collection of measurable functions \\(f\colon X\to\Bbb C\\) such that \\(\int_X|f|^p\,\rm{d}\mu<\infty\\)) can be preordered by set inclusion. In general this is not a total order. However, consider a finite measure space \\((X,\mu)\\). As a consequence of Hölder's inequality, we can actually put a total order on the \\(L^p\\) spaces with inclusion: $$ L^q(X,\mu)\subseteq L^p(X,\mu)\qquad\text{if $1\le p\le q\le \infty$.} $$

5. [#21-30, 37, 39, 40 David Tanzer](https://forum.azimuthproject.org/discussion/comment/15973/#Comment_15973):

* Any collection of sets, ordered by the inclusion relation:

+ The lattice of subspaces of a vector space, ordered by inclusion.

+ Any set of algebras of the same type, ordered by inclusion.

+ [#67 John Baez](https://forum.azimuthproject.org/discussion/comment/16085/#Comment_16085) We could even go all out and consider the collection of _all_ sets, ordered by the inclusion relation! This collection is too big to be a set. But we can get around that in various ways, e.g. by considering it as a proper class, or using a [universe](https://en.wikipedia.org/wiki/Grothendieck_universe) of "small" sets, which itself is a "large" set.

A closely related example is Ord, the class of all [ordinals](https://en.wikipedia.org/wiki/Ordinal_number). Ordinals form a _totally_ ordered class. Ord starts out like this:

$$ 0, 1, 2, 3, \dots, \omega, \omega + 1, \omega + 2, \dots, \omega \cdot 2, \dots, \omega^2, \dots, \omega^3, \dots, \omega^\omega, \dots, \epsilon_0, \dots $$

but it goes on a lot longer. In fact it goes on _longer than anything!_

* A collection of individuals, ordered by the ancestor relation. (Assuming that X is trivially an ancestor of itself.)

* Any set of propositions, ordered by the implication relation.

* Any subset of a poset is a poset.

* A Cartesian product of posets is a poset.

* We should also pay respects to the trivial poset {}.

* And every set, along with the identity ordering relation, is a poset.

* A set of formulas, ordered by the sub-formula relation, is a poset.

6. [#31 Joel Sjögren](https://forum.azimuthproject.org/discussion/comment/15988/#Comment_15988): The reflexive-transitive closure of any relation is a poset. Given A, define B inductively by (i) B(x,x) (ii) if B(x, y) and A(y, z) then B(x, z).

7. [#44 Jonatan Bergquist](https://forum.azimuthproject.org/discussion/comment/16026/#Comment_16026): the group of states in a non-reversible chemical reaction

8. [#48 Bob Haugen](https://forum.azimuthproject.org/discussion/comment/16045/#Comment_16045): a dataflow in a dataflow architecture

9. [#51 Artur Grzesiak](https://forum.azimuthproject.org/discussion/comment/16047/#Comment_16047): Let \\(P = \\{ (a,b): a < b \land (a,b) \in \mathbb{R^2} \\}\\) then following relations are posets:

1. \\((a,b)R(c,d) \iff a \le c \land b \le d \\)

2. \\(xRy \iff x \subseteq y \\)

Since this is applied course one interpretation could be that first number denotes start of some process and the second number its end. So in case of 1. a process is larger than another one if it started after another one started and ended after another one ended. While in case of 2. process is larger than another one if the latter started and ended when the first one was active.

10. [#61 Daniel Cellucci](https://forum.azimuthproject.org/discussion/comment/16071/#Comment_16071): the dependency graph of spacecraft failures, where \\(\le\\) would indicate "leads to" or "increases the likelihood of".

+ [#70 John Baez](https://forum.azimuthproject.org/discussion/comment/16091/#Comment_16091) Indeed, this kind of example is very important in applications! A [PERT chart](https://en.wikipedia.org/wiki/Program_evaluation_and_review_technique) is a way of planning tasks, where the edges indicate dependencies. There's more to a PERT chart than a mere preorder, as [Simon Willerton has explained](https://golem.ph.utexas.edu/category/2013/03/project_planning_parallel_proc.html). We may get into that later. However, any PERT chart gives rise to a preorder.

11. [#73 Andrew Ballinger](https://forum.azimuthproject.org/discussion/comment/16103/#Comment_16103): git commits appear to form a poset.

1. [#4 John Baez](https://forum.azimuthproject.org/discussion/comment/15887/#Comment_15887): the set real numbers \\(\mathbb{R}\\), with the usual notion of \\(\le\\)

2. [#5 Dan Schmidt](https://forum.azimuthproject.org/discussion/comment/15888/#Comment_15888): the set of pairs \\( (x,y) \in \mathbb R^2 \\) where \\( a \leq b \\) when \\( a_x \leq b_x \\) and \\( a_y \leq b_y \\).

3. [#15 Matthew Doty](https://forum.azimuthproject.org/discussion/comment/15951/#Comment_15951):

* Java classes form a *preordered set* when ordered by inheritance.

* Natural numbers \\(\mathbb{N}\\) form a *traditional poset* under the relation "\\(x\\) divides \\(y\\)", often denoted "\\(x\ |\ y\\)".

* For two sets \\(A,B \subseteq \mathbb{N} \\), we say \\(A\\) *is Turing reducible to* \\(B\\) if there is an [oracle machine](https://en.wikipedia.org/wiki/Oracle_machine) with oracle tape containing \\(B\\) that can compute the characteristic of \\(A\\). This is denoted \\(A \leq_T B\\). Under \\(\leq_T\\), \\(\mathcal{P}(\mathbb{N})\\) is a *preordered set*.

4. [#20 Alex Ortiz](https://forum.azimuthproject.org/discussion/comment/15965/#Comment_15965): An example of posets from analysis. If a measure space \\((X,\mu)\\) is \\(\sigma\\)-finite, then its \\(L^p\\) spaces (the collection of measurable functions \\(f\colon X\to\Bbb C\\) such that \\(\int_X|f|^p\,\rm{d}\mu<\infty\\)) can be preordered by set inclusion. In general this is not a total order. However, consider a finite measure space \\((X,\mu)\\). As a consequence of Hölder's inequality, we can actually put a total order on the \\(L^p\\) spaces with inclusion: $$ L^q(X,\mu)\subseteq L^p(X,\mu)\qquad\text{if $1\le p\le q\le \infty$.} $$

5. [#21-30, 37, 39, 40 David Tanzer](https://forum.azimuthproject.org/discussion/comment/15973/#Comment_15973):

* Any collection of sets, ordered by the inclusion relation:

+ The lattice of subspaces of a vector space, ordered by inclusion.

+ Any set of algebras of the same type, ordered by inclusion.

+ [#67 John Baez](https://forum.azimuthproject.org/discussion/comment/16085/#Comment_16085) We could even go all out and consider the collection of _all_ sets, ordered by the inclusion relation! This collection is too big to be a set. But we can get around that in various ways, e.g. by considering it as a proper class, or using a [universe](https://en.wikipedia.org/wiki/Grothendieck_universe) of "small" sets, which itself is a "large" set.

A closely related example is Ord, the class of all [ordinals](https://en.wikipedia.org/wiki/Ordinal_number). Ordinals form a _totally_ ordered class. Ord starts out like this:

$$ 0, 1, 2, 3, \dots, \omega, \omega + 1, \omega + 2, \dots, \omega \cdot 2, \dots, \omega^2, \dots, \omega^3, \dots, \omega^\omega, \dots, \epsilon_0, \dots $$

but it goes on a lot longer. In fact it goes on _longer than anything!_

* A collection of individuals, ordered by the ancestor relation. (Assuming that X is trivially an ancestor of itself.)

* Any set of propositions, ordered by the implication relation.

* Any subset of a poset is a poset.

* A Cartesian product of posets is a poset.

* We should also pay respects to the trivial poset {}.

* And every set, along with the identity ordering relation, is a poset.

* A set of formulas, ordered by the sub-formula relation, is a poset.

6. [#31 Joel Sjögren](https://forum.azimuthproject.org/discussion/comment/15988/#Comment_15988): The reflexive-transitive closure of any relation is a poset. Given A, define B inductively by (i) B(x,x) (ii) if B(x, y) and A(y, z) then B(x, z).

7. [#44 Jonatan Bergquist](https://forum.azimuthproject.org/discussion/comment/16026/#Comment_16026): the group of states in a non-reversible chemical reaction

8. [#48 Bob Haugen](https://forum.azimuthproject.org/discussion/comment/16045/#Comment_16045): a dataflow in a dataflow architecture

9. [#51 Artur Grzesiak](https://forum.azimuthproject.org/discussion/comment/16047/#Comment_16047): Let \\(P = \\{ (a,b): a < b \land (a,b) \in \mathbb{R^2} \\}\\) then following relations are posets:

1. \\((a,b)R(c,d) \iff a \le c \land b \le d \\)

2. \\(xRy \iff x \subseteq y \\)

Since this is applied course one interpretation could be that first number denotes start of some process and the second number its end. So in case of 1. a process is larger than another one if it started after another one started and ended after another one ended. While in case of 2. process is larger than another one if the latter started and ended when the first one was active.

10. [#61 Daniel Cellucci](https://forum.azimuthproject.org/discussion/comment/16071/#Comment_16071): the dependency graph of spacecraft failures, where \\(\le\\) would indicate "leads to" or "increases the likelihood of".

+ [#70 John Baez](https://forum.azimuthproject.org/discussion/comment/16091/#Comment_16091) Indeed, this kind of example is very important in applications! A [PERT chart](https://en.wikipedia.org/wiki/Program_evaluation_and_review_technique) is a way of planning tasks, where the edges indicate dependencies. There's more to a PERT chart than a mere preorder, as [Simon Willerton has explained](https://golem.ph.utexas.edu/category/2013/03/project_planning_parallel_proc.html). We may get into that later. However, any PERT chart gives rise to a preorder.

11. [#73 Andrew Ballinger](https://forum.azimuthproject.org/discussion/comment/16103/#Comment_16103): git commits appear to form a poset.