# Dictionary Definition

1 dividing an animal into right and left halves [syn: median]
2 relating to or situated in or extending toward the middle [syn: median]

# User Contributed Dictionary

## English

medial
1. Of or pertaining to a mean or average; mean
medial alligation
2. pertaining to the inside (the part facing the body); for example, the medial side of the knee faces the other knee, while the outer side of the knee is lateral.

#### Translations

of or pertaining to a mean or average
• Finnish: keski-
pertaining to the inside
• Finnish: mediaalinen, sisäpuolinen

# Extensive Definition

In abstract algebra, a medial magma (or medial groupoid) is a set with a binary operation which satisfies the identity
(x \cdot y) \cdot (u \cdot z) = (x \cdot u) \cdot (y \cdot z), or more simply, xy\cdot uz = xu\cdot yz
using the convention that juxtaposition has higher precedence. This identity has been variously called medial, abelian, alternation, transposition, bi-commutative, bisymmetric, surcommutative, entropic, etc.
Any commutative semigroup is a medial magma, and a medial magma has an identity element if and only if it is a commutative monoid. An elementary example of a nonassociative medial quasigroup can be constructed as follows: take an abelian group except the group of order 2 (written additively) and define a new operation by x * y = (− x) + (− y).
A magma M is medial if and only if its binary operation is a homomorphism from the Cartesian square M x M to M. This can easily be expressed in terms of a commutative diagram, and thus leads to the notion of a medial magma object in a category with a cartesian product. (See the discussion in auto magma object.)
If f and g are endomorphisms of a medial magma, then the mapping f.g defined by pointwise multiplication
(f\cdot g)(x) = f(x)\cdot g(x)
is itself an endomorphism.