Three-Element Groupoids

INSTRUCTIONS
Enter a 3-element groupoid in the table above.
(All entries must be either 0,1 or 2.)
Click on the "Ask Thoralf" button.
­ the groupoids that are isomorphic to the one submitted
­ the Siena Catalog number of the isomorphic copy that appears in the paper below
­ the weak isomorphism representative and its Siena Catalog number.

In the study of 3-element groupoids published in the paper mentioned below (which is called the Siena Catalog here) the groupoid tables are treated as 9-tuples (first row, second row, third row) and ordered lexicographically. From each collection of isomorphic groupoids the smallest one in the ordering was selected as the representative for the class. This gives a total of 3330 representatives (out of 19683 three-element groupoids). The 3330 representatives were then numbered from 1 to 3330, using the lex ordering, and explicitly listed in the Siena Catalog.

The main goal of the paper was to study properties of 3-element groupoids by a determination of the weak isomorphism classes. Recall that two algebras A, B are weakly isomorphic provided there is an isomorphic image C of A such that C and B generate the same clone.

This web page fills a gap in this study by listing the groupoids isomorphic to any 3-element groupoid, along with an isomorphism in each case, and it gives the Siena Catalog number of the isomorphism representative.

The last item of information that one obtains from this web page is the weak isomorphism representative of the given groupoid, a mapping that can be used to find an isomorphic copy of the given groupoid that generates the same clone as this representative, and the Siena Catalog number of this representative.

With this information one can easily use the Siena Catalog to determine key properties of the given groupoid.

Joel Berman and Stanley Burris
A Computer Study of 3-Element Groupoids
pp. 379 - 429 in Logic and Algebra
(the proceedings of the Magari conference held in Siena, Italy)
ed. by Aldo Ursini and Paolo Aglianò
Marcel Dekker, Inc., 1996.