# Quasigroups by Mark Seemann

*A brief introduction to quasigroups for object-oriented programmers.*

This article is part of a larger series about monoids, semigroups, and other group-like algebraic structures. In this article, you'll get acquainted with the concept of a quasigroup. I don't think it plays that big of a role in software design, but it *is* a thing, and I thought that I'd cover it briefly with a well known-example.

During all this talk of monoids and semigroups, you've seen that normal arithmetic operations like addition and multiplication form monoids. Perhaps you've been wondering where subtraction fits in.

Subtraction forms a quasigroup.

What's a quasigroup? It's an invertible binary operation.

### Inversion #

What does it mean for a binary operation to be invertible? It means that for any two elements `a`

and `b`

, there must exist two other elements `x`

and `y`

that turns `a`

into `b`

.

This is true for subtraction, as this FsCheck-based test demonstrates:

[Property(QuietOnSuccess = true)] public void SubtractionIsInvertible(int a, int b) { var x = a - b; var y = a + b; Assert.True(a - x == b); Assert.True(y - a == b); }

This example uses the FsCheck.Xunit glue library for xUnit.net. Notice that although FsCheck is written in F#, you can also use it from C#. This test (as well as all other tests in this article) passes.

For any `a`

and `b`

generated by FsCheck, we can calculate unique `x`

and `y`

that satisfy `a - x == b`

and `y - a == b`

.

Subtraction isn't the only invertible binary operation. In fact, addition is also invertible:

[Property(QuietOnSuccess = true)] public void AdditionIsInvertible(int a, int b) { var x = b - a; var y = b - a; Assert.True(a + x == b); Assert.True(y + a == b); Assert.Equal(x, y); }

Here I added a third assertion that demonstrates that for addition, the inversion is symmetric; `x`

and `y`

are equal.

Not only is integer addition a monoid - it's also a quasigroup. In fact, it's a group. Being associative or having identity doesn't preclude a binary operation from being a quasigroup, but these properties aren't required.

### No identity #

No identity element exists for integer subtraction. For instance, *3 - 0* is *3*, but *0 - 3* is *not 3*. Therefore, subtraction can't be a monoid.

### No associativity #

Likewise, subtraction is not an associative operation. You can easily convince yourself of that by coming up with a counter-example, such as *(3 - 2) - 1*, which is *0*, but different from *3 - (2 - 1)*, which is *2*. Therefore, it can't be a semigroup either.

### Summary #

A quasigroup is an invertible binary operation. Invertibility is the only *required* property of a quasigroup (apart from being a binary operation), but if it has other properties (like associativity), it's still a quasigroup.

I haven't had much utility from thinking about software design in terms of quasigroups, but I wanted to include it in case you were wondering how subtraction fits into all of this.

What if, however, you have a binary operation with *no other* properties?

**Next:** Magmas.

## Comments

Is division over real numbers also a quasigroup? I think so, for any number a and b we have x and y such that:

a / x = b

y / a = b

I think division over integers is not a quasigroup since for 5 and 10 there is no x such that 5 / x = 10 since 0.5 is not an integer.

Onur, thank you for writing. In general, when pondering pure mathematics rather than their programming aspects, I'm no authority. You'd be better off researching somewhere else. The Wikipedia article on quasigroups isn't too bad for that purpose, I find.

According to that article, division of non-zero rational or real numbers is an invertible binary operation.

As far as I can tell (I'm not a mathematician) you're correct that integer division doesn't form a quasigroup. When, as you suggestion, you pick

a = 5andb = 10, there's no integerxfor which5 / x = 10.