On Thursday, the Department of Mathematics hosted its first colloquium, or public lecture, of the year. The talk was well-attended by department faculty and students from a variety of disciplines who came to hear junior and math major Michael Kaminski present the research he and three other students commented this summer at Miami University in Oxford, Ohio.
The title of the colloquium was “Classifying Bol-Moufang quasigroups under a single operation.” This relates to a specific area in the field of abstract algebra. A quasigroup is a set that is closed under a binary operation—for example, if you add two numbers in the set, you get a third number that belongs to it—and satisfies the Latin square property. The Latin square property says that each element will appear exactly once in each row and column of a multiplication table.
If you have a quasigroup for which a binary operation, * (“star”) is defined, you can then define and /, or what we may call left and right division. Given a * b = c, we could left divide off a, or right divide off b. You can also define the conjugate operations and // (double left divide and double right divide), and the principle conjugate operation called “circle.” This is where the “Bol-Moufang” of the title comes in. Mathematicians Gerrit Bol and Ruth Moufang studied identities for a single binary operation and three variables, given certain ordering. Kaminski and his colleagues researched 180 such identities.
The Phillips-Vojtěchovský classification sorts these identities into a flowchart which describes which identities imply which other identities, and what is true about the quasigroup in question if a specific identity holds. Checking implications can be hard work, given that there are 180 identities. Kaminski articulated the problem thusly: “The thing is, there’s a heck of a lot of implications to check, and we don’t want to do all that work…but we still want to have a complete analysis of what implications hold and which don’t.”
Because of the way the operations work, you really only have to check implications using the * operation, because the others can be defined in terms of *. This gets you down to 60 identities whose implications need to be checked. This number can be reduced further by following the flowchart from stronger to weaker implications and stopping as soon as one fails.
Kaminski and his colleagues found all implications between identities and 52 new varieties of quasigroups of the Bol-Moufang type. They found that there are 12 minimal implications and created visual representations of their data. The application of their research is primarily to provide a reference for mathematicians who study quasigroups and other similar structures.
The talk was well received. Despite the abstract topic, Kaminski covered the subject matter comprehensively enough that a background in advanced math was not necessary to understand it. Sophomore Kasandara Sullivan said, “I think I learned something. It’s interesting to see how the shorthand simplified things. Also, left and right division is really cool.”