De Morgan Algebras. New Perspectives and Applications

It is well known that Boolean algebras can be defined using only the implication and the constant 0. It is, then, natural to ask whether De Morgan algebras can also be characterized using only a binary operation (implication) $\rightarrow$ and a constant 0. In this lecture, I give an affirmative answer to this question by showing that the variety of De Morgan algebras is term-equivalent to a (2-based) variety of type $\{\rightarrow, 0\}$. As a natural consequence, Kleene algebras can also be described as a variety using only $\rightarrow$ and 0. If time permits, I will introduce a new variety of algebras, called "Implication Groupoids", and mention some open problems. This lecture is based on the following paper: H.P. Sankappanavar, De Morgan Algebras: New Perspectives and Applications, Scientiae Mathematicae Japonicae (22 pages). To appear soon.