How to play
SEMILATTICE
SETUPYou will need . . . "Elements"
Set out a collection of colors.
In this game, you will be manipulating blobs of these colors.
The amount of each color in a blob won't be important -- the blobs in this game are stretchy.
"Equalities"Decide on some blobs to call equal (or interchangeable).
Let's allow these to be swapped out for one another.
And let's allow these to be swapped out for one another.
The colors 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 blobs equal to a particular blob, you can swap out parts of it according to the equalities you have.
Renaming (optional)Any blob 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 color equal to a blob of existing colors.
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 blobs you've discovered.
- Drag and drop pieces to and from the center area to build a blob.
- You can then select parts of this blob 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 blob. (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 sets.
When "one-way swaps" are used in place of equalities, you get a partially ordered semilattice.
| No swapping | Swapping | ||
|---|---|---|---|
 Freely making sets | One-way |  Partially ordered semilattice | |
|
Two-way
(equalities) |
 Semilattice | ||
If you start keeping track of multiple copies of a single color in a blob, you get a commutative monoid.
If you furthermore start keeping track of the order of of these pieces, you get a monoid.
If rather you keep track of order, but don't keep track of duplicate adjacent copies, you get a band.
| (With multiplicity) | |||
|---|---|---|---|
Semilattice |  Commutative monoid | ||
| (With order) |  Band |  Monoid |