How to play
COMMUTATIVE MONOID
SETUPYou will need . . . "Elements"
Set out a collection of pieces.
The game will involve manipulating bunches of copies of these pieces.
"Equalities"
Decide on some bunches to call equal (or interchangeable).
Let's allow these to be swapped out for one another.
And let's allow these all to be swapped out for one another.
The pieces and equalities you set out are just a starting point, and as the game progresses more of them will be created.
INSTRUCTIONSSwapping
From the equalities you have, you can deduce more equalities that were not already known.
To find more bunches equal to a particular bunch, you can swap out parts of it according to the equalities you have.
Renaming (optional)Any bunch can be treated as an individual.
Let's use green as shorthand for blue, yellow, red.
At any time in the game for convenience you can introduce a new piece equal to a bunch of existing pieces.
PLAY
See what equalities and pieces can be made from the ones given.
Help
- The upper panel displays the collection of pieces available in this game.
- The lower panel displays the equalities between bunches you've discovered.
- Drag and drop pieces to and from the center area to build a bunch.
- You can then select parts of this bunch with your cursor.
- Once a part is selected, anything you've already discovered equal to it will appear next to it.
- To find new equalities, swap out equal parts of a bunch. (Make selections and then click on options that pop up.)
- You can also click the pencil icon next to a selection to rename it, thereby creating a single piece to use as shorthand.
Pro tip: You can hold Control/Command or Shift to make selections with more precision.
VARIATIONS
Without equalities, it's a game of freely making bunches.
When "one-way swaps" are used instead, you get a partially ordered commutative monoid.
| No swapping | Swapping | ||
|---|---|---|---|
 Freely making bunches | One-way |  Partially ordered commutative monoid | |
|
Two-way
(equalities) |
 Commutative monoid | ||
If you stop keeping track of how many copies of each piece are in a bunch, you get a semilattice.
If you start keeping track of left-to-right placement of pieces, you get a monoid.
| No concept of multiplicity | ||
|---|---|---|
 Semilattice | Commutative monoid |  Monoid |
| No concept of order | ||
SUBTLER VARIATIONS
The ability to rename bunches is not an essential part of the game -- you could play without that feature.
Navigate by analogy| No equalities | Swapping |
Multiple arrows
(many "ways of swapping") |
|||
|---|---|---|---|---|---|
|
One-way
swaps |
Equalities
(two-way) |
One-way
arrows |
Two-way
arrows |
||
|
No renaming
(No treating bunches as individuals) |
Free commutative monoid | Partially ordered commutative monoid with chosen generators | Commutative monoid with chosen generators |  Colored prop | Colored prop with invertible arrows |
| Renaming allowed |  Partially ordered commutative monoid |  Commutative monoid |  Symmetric monoidal category | Monoidal groupoid | |