Can an algebraic structure such that $x+a=x+b$, have solutions for all $a,b∈\mathbb{K}$ exist?

Question

Does there exist an algebraic structure $(\mathbb{K},+)$ such that equations of the form $x+a=x+b$, $a\neq b$ have solutions for all $a,b\in \mathbb{K}$?

Answer 1

Let $S$ be any set. On $S$ we define $+$ the following way $$x+y=x$$

Then $(S,+)$ is a semigroup which has your property. If $S$ is infinite, your equation has infinitelly many solutions.

I think you can also add an identity to this $(S,+)$ to make it into a monoid with the required property.

Answer 2

As pointed out already in the first two comments to your question, yes, it exists.

Consider for example the finite magma with Cayley table:

$$\begin{array}{c|cccc} + & 0 & 1 & 2 & 3\\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 & 2 \\ 3 & 3 & 3 & 3 & 3 \end{array}$$

over the set $K=\{0,1,2,3\}$.

This is a right quasigroup (the operation may be defined as $x+y:=x$ for all $x,y\in K$) and, here, we have

$$x+a=x+b\quad\forall x,a,b\in K.$$