# Dictionary Definition

medial adj

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

### Adjective

medial- Of or pertaining to a mean or average; mean
- medial alligation

- 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

#### Derived terms

# Extensive Definition

- This article is about medial in mathematics. For other uses, see medial (disambiguation).

- (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.

## See also

## External links

medial in Spanish: Medial (álgebra)

# Synonyms, Antonyms and Related Words

amidships, average, banal, center, centermost, central, common, core, equatorial, equidistant, fair, fairish, halfway, indifferent, intercurrent, interior, interjacent, intermediary, intermediate, intervenient, intervening, mean, median, mediocre, mediterranean, medium, mesial, mesne, mezzo, mid, middle, middle-of-the-road,
middlemost, middling, midland, midmost, midships, midway, moderate, normal, nuclear, ordinary, routine, standard, usual