How to play
OPERAD

This game is about building structures from basic pieces.

Each piece features a collection of slots ordered from left to right, used for attaching more pieces onto it.
SETUPYou will need . . . "Elements"

Ahead of time, decide on a fixed selection of colored pieces, each with a particular number of slots.

You will be working with trees made of copies of these pieces.

"Equalities"

Decide on some trees made out of these pieces 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 trees equal to a particular tree, repeatedly swap out parts of it for their equals.

Renaming (optional)

Any tree can be treated as an individual.

Let's use green as shorthand for this complicated tree.

At any time for convenience you can introduce a new piece equal to any particular tree 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 trees you've discovered.
  • Drag and drop pieces to and from the center area to build a tree.
  • You can then select parts of this tree 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 tree. (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 trees.

When "one-way swaps" are used in place of equalities, you get a partially ordered operad.

No swapping Swapping

Freely making trees

 One-way

Partially ordered operad
Two-way
(equalities)

Operad

When all pieces have just one slot, you get a monoid.

When the slots come without left-to-right order (the input wires can cross past each other), you get a symmetric operad.

No order
on inputs 		Left-to-right order
on inputs

Symmetric operad

Planar operad
As a cosmetic choice, you might visualize the pieces as blocks with holes that fit inside each other, instead of as trees that stack on top of each other.
(Deeper nested pieces are higher up in the tree structure.)

You could make rules restricting which pieces can be slotted together the first place.

When the slots of pieces come in different kinds, and each piece can only fit into a certain kind of slot, you get a multicategory.

No restrictions Different kinds of slots
pieces have to match 		Arbitrary restrictions

Operad

Multicategory

Partial operad

Instead of a game about swapping out portions of trees, you can play similar games involving other shapes.

Line Tree ... ... ...

Monoid

Operad

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