Let (M, f) a magma. We give the link f a different name, namely
We mean that we write for f (a, b) simply
.
We can
de r name and leave out the link, when we say that it is a magma. Then you can say simply: "Let M be a magma." And anyone who knows the term "magma" know that this course includes a link.
A magma may have many additional features. We will look at a few possibilities. First, we consider
called quasigroups . A group is a quasi
Magma
fulfills the following property:
Thus, the figures
which we call left multiplication and right multiplication,
such that we each image (what is the right of the arrow) can assign exactly one prototype (what is left of the arrow).
Such pictures we call bijections.
In the integers with ordinary multiplication as the picture
assigns to each number its square, not a bijection,
because we arrange the -1 to 1, but the 1 1 in
is therefore the archetype of 1, the set {-1,1}. In a bijection the prototype of every one-element set is again a singleton.
Let Q a quasi group and
is then b = c.
This is due to the property (1) in the definition.
Conversely, for
a = c, because of Property (2).
We call this property reduction property.
A finite quasi-group is a quasi-group as the set has only finitely many elements.
A Latin square is a square of n by n fields, each field with one of n different symbols is occupied, so that each symbol in each row and each column appears exactly once each.
a link table of a quasi-group Q with k-th row and the l-th column is the product of the element that represents the
k-th row with the element that represents the l-th column.
For example, the set {0,1} with the usual addition modulo 2, a quasi-group.
addition modulo 2 means that as a result of adding only the rest take on division by 2. eg
is which means that 2 +3 is in this kind of addition first This is also evident when one sees that the 2 in this sense is a 0 (it makes the remainder is 0 mod 2) and the 3 is a 1 (she leaves the remainder 1). And
0 +1 = 1 The link table for this is
Meant with this is that 0 +0 = 0 (line 1, 1.Spalte), 0 +1 = 1 (1st line, 2.Spalte) 1 +0 = 1 (2nd line, 1.Spalte ) and 1 +1 = 0 (2nd line, 2.Spalte).
And this is obviously a Latin square. The funny thing now is that every finite group represented by a quasi-Latin square, and vice versa
represents every Latin square is a finite quasi-group.
actually has the quasi-group, even the (very complex) structure of an algebra.
But I sure would like not here, but rather to draw the eye to other important concepts.
A
left neutral element is an element e in a magma M, that is:
A right
neutral element is an element e in a magma M, such that:
A
neutral element is both left neutral, and quite neutral. has a quasi-group is a neutral element, then it is called a loop neutral element and this is a result of (1) and (2) is uniquely determined.
Let M be a magma and e is a neutral element in M.
then called a one left inverse element 
We mean that we write for f (a, b) simply 
. We can
de r name and leave out the link, when we say that it is a magma. Then you can say simply: "Let M be a magma." And anyone who knows the term "magma" know that this course includes a link. A magma may have many additional features. We will look at a few possibilities. First, we consider
called quasigroups . A group is a quasi
Magma
fulfills the following property: 
Thus, the figures
which we call left multiplication and right multiplication, such that we each image (what is the right of the arrow) can assign exactly one prototype (what is left of the arrow).
Such pictures we call bijections.
In the integers with ordinary multiplication as the picture
assigns to each number its square, not a bijection,
because we arrange the -1 to 1, but the 1 1 in
is therefore the archetype of 1, the set {-1,1}. In a bijection the prototype of every one-element set is again a singleton.
Let Q a quasi group and
is then b = c. This is due to the property (1) in the definition.
Conversely, for
a = c, because of Property (2). We call this property reduction property.
A finite quasi-group is a quasi-group as the set has only finitely many elements.
A Latin square is a square of n by n fields, each field with one of n different symbols is occupied, so that each symbol in each row and each column appears exactly once each.
a link table of a quasi-group Q with k-th row and the l-th column is the product of the element that represents the
k-th row with the element that represents the l-th column.
For example, the set {0,1} with the usual addition modulo 2, a quasi-group.
addition modulo 2 means that as a result of adding only the rest take on division by 2. eg
is which means that 2 +3 is in this kind of addition first This is also evident when one sees that the 2 in this sense is a 0 (it makes the remainder is 0 mod 2) and the 3 is a 1 (she leaves the remainder 1). And
0 +1 = 1 The link table for this is Meant with this is that 0 +0 = 0 (line 1, 1.Spalte), 0 +1 = 1 (1st line, 2.Spalte) 1 +0 = 1 (2nd line, 1.Spalte ) and 1 +1 = 0 (2nd line, 2.Spalte).
And this is obviously a Latin square. The funny thing now is that every finite group represented by a quasi-Latin square, and vice versa represents every Latin square is a finite quasi-group.
actually has the quasi-group, even the (very complex) structure of an algebra.
But I sure would like not here, but rather to draw the eye to other important concepts.
A
left neutral element is an element e in a magma M, that is:
A right
neutral element is an element e in a magma M, such that: A
neutral element is both left neutral, and quite neutral. has a quasi-group is a neutral element, then it is called a loop neutral element and this is a result of (1) and (2) is uniquely determined. Let M be a magma and e is a neutral element in M.
to b, and b is a right inverse element to a. direct from the definition of quasi-group follows that, in a loop each element is a uniquely determined left inverse and a uniquely determined right inverse element. Let M be a magma. We say that the link in this magma is
associative
applies if:
That is, the magma is associative if the order in which I run the shortcut , does not matter. One can write down
, and it is unique.
Let G be a quasi-associative group. Then G is a
group. groups are loops but also because the associativity implies together with (1) and (2) the Existence of a neutral element. Conversely, not all loops are groups, for loops need not be associative.
A group is an associative magma that has a neutral element and one in which there is a left inverse for each element. Nothing more, nothing less.
Another route from the magma to the group, therefore results on the terms semigroup and monoid. He was here briefly outlined.
Let M be a magma and the link that makes it a magma M is associative.
Then we say that M is a semigroup
.
The natural numbers with the usual addition as well as a link form a semigroup, as the natural numbers with the usual multiplication.
A semigroup with neutral element is called monoid
.
A monoid in which, for every element has a left inverse is a group.
associative
applies if:
That is, the magma is associative if the order in which I run the shortcut , does not matter. One can write down
, and it is unique.
Let G be a quasi-associative group. Then G is a group. groups are loops but also because the associativity implies together with (1) and (2) the Existence of a neutral element. Conversely, not all loops are groups, for loops need not be associative.
A group is an associative magma that has a neutral element and one in which there is a left inverse for each element. Nothing more, nothing less.
Another route from the magma to the group, therefore results on the terms semigroup and monoid. He was here briefly outlined.
Let M be a magma and the link that makes it a magma M is associative.
Then we say that M is a semigroup
.
The natural numbers with the usual addition as well as a link form a semigroup, as the natural numbers with the usual multiplication.
A semigroup with neutral element is called monoid
.
A monoid in which, for every element has a left inverse is a group.
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