Given a set $G$, we get a semigroup on $G \cup \{0\}$ as follows:
- Define $x^2 = x$ for all $x \in G \cup \{0\}$.
- Define $xy = 0$ for all distinct $x,y \in G \cup \{0\}$.
Question 0. Does this construction have a name?
Further information. The idea is that $G \cup \{0\}$ is kind of like an "internalization" of the equality relation on $G$. For example, given $g,h,i \in G$, we have that $ghi \neq 0$ in $G \cup \{0\}$ iff all three of $g,h$ and $i$ are equal.
The semigroup $G \cup \{0\}$ can also be described as the semigroup-with-$0$ presented by the generating set $G$ and the two families of relations listed above. I think its interesting that the explicit definition of $G$ "coincides" with the definition of by generators and relations. This usually doesn't happen!
There is also an order-theoretic definition of this semigroup. We think of $G$ as an antichain and adjoin a least element $0$. Then the semigroup operation on $G \cup \{0\}$ is just the order-theoretic meet with respect to the aforementioned order.
Given an algebra $\mathbf{A}=(A,F)$, the flat (one-point) extension of $\mathbf{A}$ is the algebra $\mathbf{A}^\flat=(A\cup\{0\},F\cup\{\wedge\})$ where each $f$ is extended to $A\cup\{0\}$ by setting all undefined values to 0 and $x\wedge y=0$ for distinct $x,y$ and $x\wedge x=x$. Since sets can be considered as algebras with no operations, your construction is just the flat one-point extension of a set.
See the paper "Natural dualities for semilattice-based algebras" by Davey, Jackson, Pitkethly, and Talukder for more information.