Jtdylktuy

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?

These are called magmas, not groupoids.

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

Here's three different examples.

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




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




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

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 $atimes a=a$ and $btimes b=b$ and nothing more (so $atimes b$ and $btimes a$ 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.