Quasigroup

In mathematics, especially in abstract algebra, a quasigroup is a special kind of structure that is similar to a group. The main idea is that, like in a group, you can always "divide" or reverse the operation. This means that for any two elements, there is always another element that, when combined with the first, gives the second.

Quasigroups are different from groups because they do not always need to follow the rules of being associative or having an identity element. This makes them more flexible in some ways. When a quasigroup does have an identity element, it is called a loop.

These structures are important in many areas of mathematics and help us understand how different operations can work together in various situations.

Definitions

A quasigroup is a special kind of math setup with a set of numbers and a way to combine them. Imagine you have a list of numbers, and you can always find a number to combine with another to get a specific result. This is different from regular groups because quasigroups don’t always need to follow extra rules, like having a special number that doesn’t change others when combined with them.

Quasigroups can be described in two ways: either with just one way to combine numbers, or with three special ways to combine and "undo" the combination. Both descriptions end up meaning the same thing. This makes quasigroups useful in studying patterns and solving equations in a flexible way.

Loops

A loop is a special kind of quasigroup that has an identity element. This means there is a special element, called e, such that when you combine any element x with e, you get back x. Every element in a loop also has unique partners that can "undo" it in combinations on both sides.

Loops can have weaker rules than groups, and some loops follow extra rules like the Bol loop or the Moufang loop. These loops have special patterns in how their elements combine. The term "loops" came from researchers in Chicago who were studying these structures.

Symmetries

Quasigroups can have special properties that make them more organized. One such property is called semisymmetry. In a semisymmetric quasigroup, certain rules connect the ways you can mix and divide numbers.

Another interesting property is total symmetry. In a totally symmetric quasigroup, the way you mix numbers is the same as the way you divide them. When this property also includes a rule where the order doesn’t matter, it relates to something called Steiner triples, which are special patterns of three numbers.

There is also a property called total antisymmetry, where certain rules help make sure numbers are unique in their relationships. This idea is used in some clever ways to solve problems, like in the Damm algorithm.

Examples

A quasigroup is a special kind of math structure. One example is the set of all whole numbers with subtraction. Even though this set acts like a quasigroup, it is not a loop because it lacks a special element that works like a "starting point" in both directions.

Another example is the set of nonzero rational or real numbers using division. This also forms a quasigroup. There are more complex examples, like certain sets of numbers used in advanced math, which create loops that are not groups. These show how quasigroups can be different from the more familiar groups in math.

Properties

Quasigroups have a special rule called the cancellation property. This means that if you multiply two numbers and get the same result as multiplying another number by the same amount, you can be sure the two numbers are actually the same.

Quasigroups also have something called the Latin square property. This means that if you know any two numbers in a multiplication problem, you can always figure out the third number. This makes quasigroups very organized and predictable.

Main article: Latin square

Morphisms

A quasigroup homomorphism is a special kind of mapping between two quasigroups that keeps their structure the same. It means that when you apply the homomorphism to the results of combining two elements, it's the same as combining the results of applying the homomorphism to each element separately.

Homotopy and isotopy

Main article: Isotopy of loops

In quasigroups, a homotopy is a way to connect two quasigroups using three special mappings. If these mappings are all the same, it's called a quasigroup homomorphism. An isotopy is a special type of homotopy where each mapping is a perfect one-to-one match. When two quasigroups have an isotopy between them, they are called isotopic. This idea relates to changing rows, columns, and elements in a square table of values.

Conjugation (parastrophe)

Quasigroups can have their operations changed in six different ways by switching the order of elements or using division. These changed operations are called conjugates or parastrophes of the original operation.

Isostrophe (paratopy)

When two operations in a quasigroup are related through isotopy and switching, they are called isostrophic or paratopic to each other.

Generalizations

Polyadic or multiary quasigroups

An n-ary quasigroup is a set with an n-ary operation, (Q, f) with f : Qⁿ → Q, such that the equation f(x₁, ..., x_n) = y has a unique solution for any one variable if all the other n variables are specified arbitrarily. Polyadic or multiary means n-ary for some nonnegative integer n.

A 0-ary, or nullary, quasigroup is just a constant element of Q. A 1-ary, or unary, quasigroup is a bijection of Q to itself. A binary, or 2-ary, quasigroup is an ordinary quasigroup.

An example of a multiary quasigroup is an iterated group operation, y = x₁ · x₂ · ··· · x_n; it is not necessary to use parentheses to specify the order of operations because the group is associative. One can also form a multiary quasigroup by carrying out any sequence of the same or different group or quasigroup operations, if the order of operations is specified.

There exist multiary quasigroups that cannot be represented in any of these ways. An n-ary quasigroup is irreducible if its operation cannot be factored into the composition of two operations in the following way:

f(x₁, ..., x_n) = g(x₁, ..., x_i−1, h(x_i, ..., x_j), x_j+1, ..., x_n),

where 1 ≤ i 2; see Akivis & Goldberg (2001) for details.

An n-ary quasigroup with an n-ary version of associativity is called an n-ary group.

Number of small quasigroups and loops

Main article: Small Latin squares and quasigroups

This section shows the number of different small quasigroups and loops. These are special math patterns that are counted in a list found online.