Combinator Algebra and Unary Monoids

The subject of combinator algebra concerns simple algebraic structures intended to abstract the operation of function application, and has deep connections to logic and computer science. A unary monoid is a monoid equipped with a unary operation, which need not satisfy any additional axioms. For example, any group or restriction monoid is a unary monoid. In any unary monoid we can define an "application" operation and study it from the perspective of combinator algebra. In this talk I will give an introduction to combinator algebra and explore the connection to specific unary moniods. The highlight is likely a charactersation of the boolean groups obtained in this way. No background beyond rudimentary abstract algebra is required.