# What is an example of a groupoid which is not a semigroup?

I know that groupoid refers to an algebraic structure with a binary operation. The only necessary condition is closure.

However, I couldn't find any easy-to-understand example of a groupoid which is not a semigroup. I did come across some examples of (certain type of) matrices but then matrix multiplication is always associative (thus making it a semi-group).

So, could someone please provide me an example of a groupoid which isn't a semigroup?

• You should be careful with the word "groupoid". In the 60s some people used this word to refer to "sets with a binary operation" (possibly non-associative), but these are now called magmas. Groupoids, on the other hand, are sets (or classes) with a partially defined binary operation satisfying group-like properties. – Luiz Cordeiro May 19 '18 at 17:50
• @LuizCordeiro Thanks. I didn't know that. However, I guess in the context of this question: "Give an example of a groupoid which is not a semigroup." I think they mean "magma" by "groupoid"? – user563280 May 19 '18 at 17:57
• Yes, since you're looking for an example of a non-associative binary operation, you're looking for a magma which isn't a semigroup. – Luiz Cordeiro May 19 '18 at 18:01

These are called magmas, not groupoids.

The midpoint'' operation $s\ast t=\frac{s+t}{2}$ on $\mathbb{R}$ makes it a magma which is not a semigroup.

• s*t = s-t => Is also a groupoid but not a semi-group. I learned that these are being called magmas now. Thanks! – router Dec 18 '18 at 4:33

Here's three different examples.

1. Take an abelian group $(A,+)$ and define a new binary operation $\circ$ on $A$ by $x\circ y=x+(-y)$. This is an example of a quasigroup.

2. Take a group $(G,\cdot)$ and define a new binary operation $\triangleleft$ on $G$ by $x\triangleleft y=x\cdot y \cdot x^{-1}$. This is an example of a quandle.

3. Take a digraph $(V,A)$ with the property that for any two distinct vertices $v,w\in V$, exactly one of the arcs $vw$ or $wv$ is in $A$. Define a commutative binary operation $\cdot$ on $V$ by $v\cdot v=v$ and $v\cdot w=w$ if and only if $vw\in A$. This is an example of a tournament.

A quasigroup is associative if and only if it is a group, a quandle is associative if and only if it is trivial, and a tournament is associative if and only if it is a commutative idempotent semigroup (aka a semilattice).

$$( \Bbb Z , -)$$

This is a groupoid and not a semi-group.

Let $a,b,c$ be distinct members of a three element set and $ab=c=cc$, $bc=a=aa$ and define $ac, bb$ however you like (but in $\{a,b,c\}$.) You have a nonassociative binary operation.

Let $\{a,b\}$ be a set with two distinct elements.

Define a partial multiplication by $a\times a=a$ and $b\times b=b$ and nothing more (so $a\times b$ and $b\times a$ are not defined).

Then $\{a,b\}$ equipped with this multiplication is a groupoid, but not a semigroup.

I suspect you use another definition of groupoid. If it is what I would call a magma then see the other answers.

First, the term 'groupoid' recently rather means primarily a category with invertible arrows, and the term 'magma' is arising for an algebraic structure with a binary operation.

For a familiar example, consider $\Bbb Z$ (or almost any Abelian group) with the subtraction.

Or, define $a*b:=a+b+1$ (or whatever..)

Other examples arise e.g. from finite quasigroups whose multiplication table is a latin square: having each element once in every row and in every column.