Special Element in a Magma Say I have a Magma $(M,\times)$, on which there exists an element $k\in M$ such that $k\times x = x\times k = k, \forall x\in M$. $k$ is then called an absorbing element (or zero element). 
My question is, say we have another magma $(G,\cdot)$. Say there exists an element $s\in G$ and an element $t\in G$ such that $s\neq t$ and $s\cdot x = x\cdot s = t, \forall x\in G$, does this sort of concept make sense? If so, does it have a name?
Much appreciated.
 A: Say we have a magma $(G,\cdot)$. Say there exists an element $s\in G$ and an element $t\in G$ such that $s\neq t$ and
$$s\cdot x = t = x\cdot s\quad \forall\,x\in G.$$
I don't think such situation or couple of elements have a name in particular but this is how the Cayley table looks like:
$$\begin{array}{c|cccccccc}
\cdot & 0 & 1 & 2 & s & 4 & t & 6 & \ldots\\ \hline
0 & & & & t & & & & \\
1 & & & & t & & & & \\
2 & & & & t & & & & \\
s & t & t & t & t & t & t & t & \ldots\\
4 & & & & t & & & & \\
t & & & & t & & & & \\
6 & & & & t & & & & \\
\vdots & & & & \vdots & & & & \\
\end{array}$$
May such groupoid have an absorbing element too? Yes, but $t$ only: if
$$t\cdot x = t = x\cdot t\quad \forall\,x\in G.$$
is verified also, we have
$$\begin{array}{c|cccccccc}
\cdot & 0 & 1 & 2 & s & 4 & t & 6 & \ldots\\ \hline
0 & & & & t & & t & & \\
1 & & & & t & & t & & \\
2 & & & & t & & t & & \\
s & t & t & t & t & t & t & t & \ldots\\
4 & & & & t & & t & & \\
t & t & t & t & t & t & t & t & \ldots\\
6 & & & & t & & t & & \\
\vdots & & & & \vdots & & \vdots & & \\
\end{array}$$
I always think in terms of finite structures first: in this case this fits. Structures over non-finite sets have no Cayley table instead, but this may give you a hand understanding what the identities in your question mean in general.
