We will investigate algebraic structures that are isomorphic to a set of partial functions equipped with selected operations (e.g. composition, intersection, etc.). I will spend most of the time describing results about completely representable algebras - these are the algebras that can be embedded in a set of partial functions in a way that preserves arbitrary cardinality meets and/or joins. Specifically, I will show how we can axiomatise this subclass of algebras (for various choices of operations) [1].

I will also state, but not prove, some other recent results about algebras of partial functions: expressivity results about the choice of operations [2], and also possibly one or two duality results for classes of these algebras [3,4].

[1] B. McLean, Complete representation by partial functions for signatures containing antidomain restriction, International Journal of Algebra and Computation (to appear), 2024.
[2] B. Bogaerts, B. ten Cate, J. Van den Bussche, and B. McLean, Preservation theorems for Tarski's relation algebra, Logical Methods in Computer Science (to appear), 2024.
[3] B. McLean, A categorical duality for algebras of partial functions, Journal of Pure and Applied Algebra 225 (2021), no. 11, 106755.
[4] C. Borlido, B. McLean, Difference-restriction algebras of partial functions with operators: discrete duality and completion, Journal of Algebra 604 (2022), 760-789.