Mastermind for Monoids

The monoid of integers modulo three Z/3Z wth addition as operation is an example of a monoid of small order. The multiplication table can help to see if there are other monoids of order three.

Two important observations are helpful.

• The existence of a unit implies that one row and one column are already determined.
• Due to the cancellation property a row or column underneath an invertible element cannot contain an element twice.

1. Try to find another monoid of order 3, that is not isomorphic to Z/nZ, or to set up a reasoning why this cannot be the case.
2. How does this work for monoids of order 4?

Note that you can use

• exchanging rows or columns to reorder the elements (which does not change the structure, of course);
• coloring of the elements to see the isomorphism easily.
• Check that your structure is indeed a monoid (or semi-group, or group, or just associative) by clicking on the green square in the upper left corner of the multiplication table.