Groupoid: Frequently asked questions
Question 1. Why only 62 elements?
Answer. This is just a small program with no ambition to do everything. Often we just need to quickly check if a small groupoid is subdirectly irreducible or if it satisfies a certain equation. Then speed of input may be the most important factor. To be able to quickly input the multiplication table, it is desirable that every element is assigned a label consisting of a single character. A natural choice are the 62 characters a,..,z,A,..,Z,0,..,9. If elements were to be labeled by lengthier expressions, the multiplication tables would become much bigger.
Also, most finite groupoids have at most 62 elements, or even less. In the past several years the largest groupoid that I ever needed to input for this program had 21 elements.
Somebody (was it V. I. Lenin?) said: "First we need to understand small groupoids and only after that can we tackle the larger ones."
Question 2. What if I need a groupoid with 63 elements?
Answer. If you really need such a monster, try to download the Algebra Calculating Program by R. Freese and E. W. Kiss.
Question 3. Up to 62 is O.K., but I need two binary operations. Or at least one unary operation together with the multiplication. What should I do?
Answer. Try to download the Algebra Calculating Program
Question 4. I need to check an algebra for subdirect irreducibility. But the signature is one binary symbol plus a constant.
Answer. Forget about the constant. Constants play no role in congruence generation.
Question 5. It crashed.
Answer. Yes, it happens. It doesn't happen often if you just sit and do not press any key or do not click with the mouse. Or, try to rerun the program after it crashes; perhaps next time you will be more lucky.
Question 6. If it crashes, should I report to Microsoft and ask them to debug the program?
Answer. Please don't. If you get such an idea, better contact me at jezek@karlin.mff.cuni.cz
Question 7. The messages go out of the screen to the right when I have more than 5 groupoids. I can't read them. Should I buy a larger monitor?
Answer. You don't need to. Try to drag the screen to the left with the left mouse button pressed.
Question 8. Saving doesn't work! I tried to save it, didn't get any error message, and the file was not there!
Answer. There are groupoids and equations to be saved. If you want to save the equations, the filename's extension must be 'e'. If you want to save the algebra (only the last one will be saved), the extension must be 'a'. Nothing is saved if the extension is anything else.
Question 9. It is so unreliable! I worked with several groupoids and then when I wanted to save them to a file, only the last one was saved.
Answer. Save the last, delete it, again save the last, delete it, etc.
Question 10. Can I input a groupoid so that its zero is 0 and the unit is 1?
Answer. Only if the groupoid has between 54 and 62 elements.
Question 11. I needed to check a certain groupoid for subdirect irreducibility. I tried to hit each key over its title field and I also clicked all buttons, but there doesn't seem to be this service.
Answer. Hit the key 'i'. A groupoid is subdirectly irreducible if and only if it has just one minimal congruence.
Question 12. Why must the mouse point to the groupoid's title field when I need to press 'i' to get the list of minimal congruences? It is so difficult to position the mouse precisely there and then, when I release the mouse to hit the key, it moves away from the field. Why is it not sufficient to position it anywhere over the multiplication table? The title field is so narrow.
Answer. If you press 'i' when the mouse points inside the multiplication table, then (if the groupoid has at least 9 elements) the program assumes that you want to correct one of the products in the table to give it this value i.
Question 13. What is it good for?
Answer. Universal algebras and groupoids, in particular, can be applied in many various branches of human activity. Because the number of applications is so large, let me select just one example: an application to the environment in the country around a dam is given in the paper by O. Hasik: Matematicke modely interakci vodnich nadrzi v zivotnim prostredi, published in Vodni hospodarstvi, rada A, 2/1982, 47-54.
Question 14. I like this program because at work when anybody goes around I can easily jump to it from SuperShoots which I like to play, and Groupoid looks quite professional. Only that sometimes I must explain them what is it about, and there is my weak place. What do you mean by a groupoid?
Answer. A groupoid (sometimes also called binary system) is a nonempty set with one binary operation.
Question 15. Groupoid is not a binary system! Groupoid is a small category in which every morphism is an isomorphism. Even the AMS 2000 Mathematics Subject Classification recognizes groupoids as categories.
Answer. Nevertheless, there are hundreds, perhaps thousands of papers on groupoids in the sense of binary systems, while there are only tens of papers on groupoids meant as these structures between groups and categories.
Question 16. I would write it better.
Answer. I am sure you would.