Enter a 3-element groupoid in the table above.
must be either 0,1 or 2.)
Click on the "Ask Thoralf" button.
You will receive:
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