3 problems

Let $\mathbf{M}$ denote the class of metric spaces, let $\mathbf{F}$ be the monoid of functions under composition, and let $\mathbf{P}_{\mathbf{X}}$ be the monoid associated with a…

Let $\mathbf{M}$ denote the class of metric spaces, let $\mathbf{P}_{\mathbf{X}}$ be the monoid associated with a metric space $\mathbf{X}$, and let $\mathbf{Am}$ be the monoid of…

Let $f: [0,\infty) \to [0,\infty)$ be a function. Let $\mathbf{A}$ be the class of all unbounded boundedly compact (proper) ultrametric spaces, and let $\mathbf{P}_{\mathbf{A}}$ de…