The symmetries of a geometric shape \(X\) are the distance preserving functions \(f:X \rightarrow X\). Of course, the concept of symmetry extends well beyond Geometry and into virtually every area of Mathematics. For example, if \(X\) is a model-theoretic structure or a topological space, then its symmetries are the automorphisms and homeomorphisms of \(X\), respectively. In each case, the symmetries of \(X\) form a subgroup of the symmetric group \(\operatorname {Sym}(X)\) of all permutations of \(X\).

There are two widely-studied and natural generalisations of the group \(\operatorname {Sym}(X)\) to the world of semigroups. The Full Transformation Monoid \(X^X\) consists of all functions \(f:X\rightarrow X\) and the Symmetric Inverse Monoid consists of all bijections between subsets of \(X\). These two semigroups correspond to two generalisations of the concept of symmetry as described above. For example, the group of automorphisms of a structure \(X\) is a subgroup of the semigroup of homomorphisms \(f:X\rightarrow X\) (which is a subsemigroup of \(X^X\)) and of the inverse semigroup of all isomorphisms between substructures of \(X\) (which is an inverse subsemigroup of \(I_X\)).

In this talk, we will consider \(\operatorname {Sym}(X)\), \(X^X\), and \(I_X\) on an infinite set \(X\). I will present a variety of results which highlight the connections between Semigroups on the one hand and Set Theory, Model Theory, and Topology on the other. No previous knowledge of Semigroup Theory will be assumed.