The injective continuous map $h:X\to Y$ is a topological embedding if it is a homeomorphism onto the image of $h$ in $Y$.
Let the function $h:X\to Y$ be a topological embedding and let $X=Y$ be a semiring.
Does this have a special name or notable properties in respect of either a) being a topological embedding in itself and b) being a topological emmbedding in a semiring?
Motivation
I don't believe this adds anything to the question tbh. but I was advised to add it. I'm making slow progress proving the following:
Let $T$ be the unique homeomorphism on the 2-adic numbers which topologically conjugates $f(x)=3x+2^{\nu_2(x)}$ to $g(x)=x-2^{\nu_2(x)}$
Let $X$ be the semiring of positive ternary rationals as a subspace of $\Bbb Z_2$ . I want to show that $T$ is a topological embedding of $X$ in itself.
but I'm just working from scratch algebraically treating $X$ as a set (susbset of $\Bbb R$), not using the concepts of topological embedding, nor the semiring other than closure under addition and multiplication, and I wanted to see if there's any material on how this concept (of a topological embedding on a semiring) can be of help, if at all.