An Algebraic and Logical approach to continuous images

Continuous mappings between compact Hausdorff spaces can be studied using homomorphisms between algebraic structures (lattices, Boolean algebras) associated with the spaces. This gives us more tools with which to tackle problems about these continuous mappings -- also tools from Model Theory. We illustrate by showing that the \v{C}ech-Stone remainder $[0,\infty)$ has a universality property akin to that of $N^*$; a theorem of Ma\'ckowiak and Tymchatyn implies it own generalization to non-metric continua; and certain concrete compact spaces need not be continuous images of $N^*$.