Someone I know is trying to figure out if the following concepts already have an established name in the literature, and MO is a great place to ask around.
1) Suppose $X$ is a metric space equipped with an associative product and a unit element. Let $m: X \times X \to X$ be the product. Suppose also that $m$ is nonexpansive, i.e. $$ d\big( m(x,x'),m(y,y') \big) \le d\big((x,x'),(y,y')\big) $$ say when $X\times X$ is given the Euclidean metric with respect to the given metric of $X$. Is there a standard name for this type of structure?
2) Let $f : X \to Y$ be a map between metric spaces with the property that for all $x,y \in X$ $$ d(f(x), f(y)) \geq d(x,y). $$ So in a way this is the opposite of a nonexpansive map. Is there a name for such maps?