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Given a set $G$, we get a semigroup on $G \cup \{0\}$ as follows:

  1. Define $x^2 = x$ for all $x \in G \cup \{0\}$.
  2. 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.

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  • $\begingroup$ It is a special case of idempotent and commutative semigroup (or semilattice). To my knowledge, it does not have a special name in semigroup theory, but it may have a name in lattice theory. $\endgroup$
    – J.-E. Pin
    Apr 13, 2015 at 6:11
  • $\begingroup$ @J.-E.Pin, sure, but I already knew that. See the last paragraph :) Edit. But thanks for the information that it probably doesn't have a name in semigroup theory. $\endgroup$ Apr 13, 2015 at 6:12
  • $\begingroup$ In any partial ordered set with a bottom element, the meet-semilattice of atoms (en.wikipedia.org/wiki/Atom_(order_theory)) is a well-defined semilattice. So I would call it just this: "meet-semilattice of atoms". $\endgroup$ Apr 13, 2015 at 8:57
  • $\begingroup$ @ThomasKlimpel, better to say: "meet-semilattice of subatoms," where a subatom of $(P,\bot)$ is defined as an $x \in P$ such that if $a < x$, then $a= \bot.$ It follows trivially that $x \in P$ is a subatom iff it is either an atom, or $\bot$. $\endgroup$ Apr 13, 2015 at 12:09
  • $\begingroup$ If you introduce the name "subatom" for an element which is either the bottom or an "atom", then you can no longer pretend that you are using established terminology. If you slightly abuse existing terminology on the other hand, then you are just following established mathematical practice. (red herring principle...) $\endgroup$ Apr 13, 2015 at 14:18

1 Answer 1

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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.

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  • $\begingroup$ This is a perfect answer. $\endgroup$ Apr 16, 2015 at 6:50

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