Matthew Ward, an undergraduate Mathematics major, won the "Best Poster" award at the Undergraduate Poster Session in New Orleans, LA for his poster entitled " An initial inquiry into LC loops." The competition was held as part of the national 2007 Joint Mathematics Meeting of the AMS, MAA, and SIAM.
Matt's poster examines the structural properties of loops that satisfy the LC property: (xx)(yz)=(x(xy))z, a weakened association.
A loop is a nonassociative group. LC-loops are one of two types of Bol-Moufang loops that have not been studied previously in any depth. The other type that hasn't been studied is known as RC. This is just the dual of the LC property, so any structural result gained from LC-loops can be immediately applied to RC-loops. Properties of the nucleus, order of the elements, the scre property, and simple loops are discussed. The nucleus refers to the set of elements that associate in a very limited and specific way. It turns out that any LC-loop of odd order is actually a group, so emphasis is put on the study of even ordered LC-loops. Imposing the scre property on LC-loops returns many powerful results. The scre property is a weak form of commutativity. It states that every element that is a square commutes with all other elements of the loop. One of the strongest structural theorems we proved was that any LC-loop of order 2p where p is an odd prime is simple. The ultimate goal is to completely classify simple LC-loops and to come up with a complete decomposition theorem.
Matt wrote an accompanying paper entitled "An Initial Inquiry into LC-loops", which has been accepted for publication in the American Journal of Undergraduate Research.