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Encyclopedia > Auto magma object

In mathematics, a magma in a category, or magma object, can be defined in a category with a cartesian product. This is the 'internal' form of definition of a binary operation in a category.

As Mag the magma category hasdirect products, the concept of an (internal) magma (or internal binary operation) in Mag is defined, say

T′: (X,T) � (X,T) → (X,T).

Since T′ is a morphism we must have

(x Ty) T (u Tz) = (x T u) T′ (y T z).

If we want to take the original operation, this will be allowed only if the medial identity

(x T y) T (u T z) = (x T u) T (y T z)

is valid.

This operation, which gives a medial magma, can have a two-sided identity only if it is a commutative monoidal operation. The if direction is obvious.

As a result Med, the medial category, has all its objects as medial objects; and this characterizes it. Results from FactBites:

 Magma (algebra) - Wikipedia, the free encyclopedia (535 words) The term magma for this kind of structure was introduced by Bourbaki; however, the term groupoid is a very common alternative. Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation. A free magma on a set X is the "most general possible" magma generated by the set X (that is there are no relations or axioms imposed on the generators; see free object).
 Printing (1727 words) For "object", the corresponding object of the expression arguments is printed in a default printing mode, while for "magma" the corresponding object of the expression arguments is printed in Magma printing mode. In the file object case, the string obtained from the formatted printing of the other arguments is simply appended to the file. Magma allows the user to obtain the string corresponding to the output obtained when printing an object by means of the
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