Asha Gair is a mathematics major at La Trobe University heading into the honours component of her Bachelor of Science in 2015. She also has a major in statistics. Asha has previously completed a Bachelor of Biological Science, with major in genetics and biochemistry, where she took mathematics as her electives purely for fun. She then returned to study statistics to improve her employment prospects and continued to study mathematics for fun, but, in the process, has found that mathematics is where her heart lies.
Her passions are in general algebra, lattice theory and duality theory; areas she hopes to expand her knowledge of during the AMSI Vacation Research Scholarship and further research. While she is drawn to pure mathematics, she also has a love for using mathematics to understand the universe through cosmology and astrophysics. Outside of her time at La Trobe, Asha is an avid reader of everything from fantasy novels and comic books to scientific texts. She also enjoys spending her limited free time playing video and board games with friends. Most recently she has also developed a love of costume making.
Quasi-primal Cornish Algebras
The two-element Boolean algebra plays a hidden but fundamental role in our everyday lives, forming the logical basis of electronic circuits and computers. What makes this algebra so special? Every first-year computer science student learns how this algebra can be used to build all possible circuits (more formally, all possible operations on the set f0; 1g). Researchers in universal algebra found that one particular operation (called the ternary discriminator) gives the two-element Boolean algebra many of its special properties. This led them to generalise the two-element Boolean algebra to the much wider class of quasi-primal algebras.Quasi-primal algebras are of ongoing interest among researchers working on the applications of algebra to the study of logic. They also arise naturally in classical algebra (every finite field is quasi-primal) and play an important role in universal algebra (for example, in McKenzie and Valeriote’s ground-breaking characterisation of varieties with a decidable theory). Recently, Davey, Nguyen and Pitkethly characterised the quasi-primal Ockham algebras. Ockham algebras are the algebraic counterpart of the non-classical logic in which the law of the excluded middle and the double negation law are dropped but the De Morgan laws are retained. Cornish introduced a natural generalisation of Ockham algebras, which are now named after him. Cornish algebras are the algebraic counterpart of certain non-classical logics with more than one concept of negation. It is natural to seek an extension to Cornish algebras of the characterisation of quasi-primal Ockham algebras.