Cody Allen stands next to his poster, “Arithmetic Congruence Monoids and Full Elasticity,” at SDSU’s 6th annual Student Research Symposium. Cody worked with three other students last summer in SDSU’s Research Experience for Undergraduates program (REU), led by Dr. Vadim Ponomarenko. Cody is now studying for his graduate degree in Applied Mathematics and hopes to continue on to obtain a PhD.
“An arithmetic congruence monoid (ACM) is an arithmetic progression defined by integers a and b such that both a and b are positive satisfying the following conditions: a is less than or equal to b and a squared is congruent to a modulo b. This algebraic structure is a non-unique factorization domain, which means elements may have more than one distinct factorization. Elasticity of an element, say g, is defined as the longest length factorization of g divided by the shortest length factorization of g. Our specific research was regarding full elasticity, which is satisfied by the following condition: an ACM is fully elastic if for every rational number h, where h is greater than or equal to one and less than or equal to the elasticity of the monoid defined by a and b, then there exists an element h’ such that the elasticity of h’ is equal to h.
Our approach was to construct sub-monoids where we could fully characterize elements and then show full elasticity using only that subset of elements. This process relied on taking specific prime numbers p and q and then generating a product PQ, where the number of copies of p and the number of copies of q in the element PQ varied. These elements, PQ are the elements that allowed us to construct the particular rational numbers required to satisfy the full elasticity requirement.
The summer REU program hosted by SDSU and led by Dr. Vadim Ponomarenko provides students with the opportunity to investigate different areas of mathematics and feel the excitement of conducting mathematical research.”
Cody’s poster may be seen in greater detail at: http://www-rohan.sdsu.edu/~allen1/research.html