Magma (algebra)

In abstract algebra, a magma is one of the simplest types of algebraic structure. It starts with a set of elements — think of it as a collection of different items. What makes a magma special is that it has one way to combine any two of these items, called a binary operation. This operation must always give another item from the same set, which we call being closed.

Even though magmas are basic, they are important because they form the foundation for more complex structures. For example, if you add just one extra rule — like the operation being associative — you get a semigroup. Add another rule, and you might end up with a group, which is a key idea in many areas of mathematics.

The term “magma” was introduced by the mathematician René Thuillier in the 1930s, and it helps mathematicians study how different rules for combining things change the overall behavior of a structure. Though simple, magmas show us how even the most basic rules can lead to rich and varied mathematical worlds. For related ideas in category theory, see Groupoid, and for other uses of the word “magma,” see Magma (disambiguation).

History and terminology

The word groupoid was first used in 1927 by a mathematician named Heinrich Brandt. Later, other mathematicians used the word in a different way. This caused some confusion because groupoid now means different things to different mathematicians.

Some writers today use the word magma instead. This term was suggested by another mathematician, Serre, in 1965. It is used when talking about a set with a special kind of calculation that doesn’t have to follow the usual rules, like addition or multiplication.

Definition

A magma is a special kind of mathematical structure. It has a group of items, called a set, and a way to combine any two of these items. This combining method is called an operation. When we take any two items from the set and use this operation, the result is always another item from the same set. This is called the closure property.

If the operation can only be used for some pairs of items and not others, then the structure is called a partial magma or a partial groupoid.

Morphism of magmas

A morphism of magmas is a special kind of function that connects two magmas while keeping their operations consistent. For example, think of one magma made from positive real numbers using the geometric mean, and another magma made from all real numbers using the arithmetic mean. A logarithm can act as a morphism, linking these two magmas.

This idea has been useful in economics. A scientist named W. Stanley Jevons used it to study changes in prices, or inflation, of different goods in England.

Notation and combinatorics

In algebra, when we do operations in a magma many times, the order can change the result. We use parentheses to show the order, like ((a • (b • c)) • d). Sometimes we can make this easier to read by leaving out some parentheses, like writing xy • z instead of (x • y) • z.

There are special ways to write these operations without using many parentheses, such as prefix notation (••a•bcd) or postfix notation (abc••d•). The different ways to arrange operations for n steps are counted by the Catalan numbers. For example, with three steps, there are five different ways to arrange the operations.

We can also count how many different magmas exist with a certain number of elements. For example, there are 16 different magmas with 2 elements, and many more as the number of elements grows.

Free magma

A free magma M_X on a set X is the simplest magma you can make from X. It has no extra rules or limits. The operation joins two items by putting them in parentheses in order. For example, a combined with b becomes (a)(b).

You can think of a free magma like building shapes with full binary trees, where the leaves are labelled by elements of X. The operation is like connecting these trees at their roots.

Types of magma

Magmas are special sets with a rule for combining two elements. There are many types of magmas depending on extra rules we can add. Some common types include:

• Quasigroup: A magma where you can always "divide" elements.
  • Loop: A quasigroup with a special element that doesn’t change others when combined with them.
• Semigroup: A magma where the combination rule works in a chain-like way.
  • Monoid: A semigroup with a special element that doesn’t change others when combined with them.
• Group: A magma with a special element, chain-like combinations, and the ability to "undo" any combination.

We can also have magmas where the order of elements doesn’t matter:

• Commutative magma: A magma where combining in any order gives the same result.
• Commutative monoid: A monoid where the order doesn’t matter.
• Abelian group: A group where the order doesn’t matter.

Classification by properties

A magma is a special math idea made from a group of items and a way to combine any two of them. This combination doesn’t need to follow extra rules, but mathematicians give names to magmas that do follow certain rules.

For example, a magma is called commutative if combining x and y gives the same result as combining y and x. It is idempotent if combining an item with itself just gives back that same item. Some magmas are associative, meaning the way you group the combinations doesn’t change the result. There are many other special names for magmas with different properties.

Number of magmas satisfying given properties

In algebra, a magma is a group of items with one way to connect any two items together. This connecting doesn't need to follow special rules, like matching up in order or grouping together. This makes magmas a simple starting point for studying more complicated math ideas.

The number of different magmas we can make with a certain number of items depends on how many ways we can define the connection between those items. For a small group, we can count these possibilities. This helps mathematicians see how more complex math systems can grow from simpler ones.

Category of magmas

The category of magmas is called Mag. It includes all magmas as its objects and special maps called magma homomorphisms as connections between them. This category has direct products and can include simple sets as special kinds of magmas.

One important feature is that some maps within a magma can be expanded to full transformations of the magma, like finishing a sequence of steps.