A link table is a table with which two-digit links are shown in mathematics and especially in algebra . For example, the following table shows the multiplication on the quantity : ${\ displaystyle \ cdot: \ \ mathbb {Z} _ {2} \ times \ mathbb {Z} _ {2} \ to \ mathbb {Z} _ {2}}$${\ displaystyle \ mathbb {\ mathbb {Z}} _ {2} = \ {- 1,1 \}}$

${\ displaystyle \ cdot}$ 1 −1
1 1 −1
−1 −1 1

Link tables appear in propositional logic in the form of truth tables . In group theory , they can be used to write down or construct (small) groups .

The display as a link table is suitable for any link . Such a link associates each pair of elements and one element . This assignment can be shown in a table as follows: ${\ displaystyle \ circ \ colon A \ times B \ to C}$${\ displaystyle c = a \ circ b}$${\ displaystyle a \ in A}$${\ displaystyle b \ in B}$${\ displaystyle c \ in C}$

${\ displaystyle \ circ}$ ${\ displaystyle \ dots}$ ${\ displaystyle b}$ ${\ displaystyle \ dots}$
${\ displaystyle \ vdots}$
${\ displaystyle a}$ ${\ displaystyle a \ circ b}$
${\ displaystyle \ vdots}$

The first argument is in the input column, the second argument is in the header , and the result of the link is found at the intersection of -line and -column . ${\ displaystyle a \ in A}$${\ displaystyle b \ in B}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle c = a \ circ b}$

In order to be able to write down the table in full, one also assumes that the sets and are finite, and also sufficiently small for practical purposes. ${\ displaystyle A}$${\ displaystyle B}$

Often link tables are used for internal links (i.e. in the case ) and here in particular for groups. ${\ displaystyle A = B = C}$

## Examples

### Examples from logic

Truth tables used in the propositional logic to the result of the logical operations ( logical connectives to be described) or to define. Three typical examples are

• the conjunctor (logical "and"),${\ displaystyle \ wedge}$
• the disjunctor (logical "or"),${\ displaystyle \ vee}$
• the implication (logical "if ... then ...").${\ displaystyle \ Rightarrow}$

The following tables show the linkage tables for these junctions:

${\ displaystyle \ wedge}$ true not correct
true true not correct
not correct not correct not correct
${\ displaystyle \ vee}$ true not correct
true true true
not correct true not correct
${\ displaystyle \ Rightarrow}$ true not correct
true true not correct
not correct true true

The first two tables are immediately obvious. The third, on the other hand, is less intuitive: It expresses the fact that correct inference from true assumptions can only lead to true conclusions (first line), but that one can draw both false and true inferences from false assumptions (second line). This example shows that the logical connections also require a clarifying definition, and the truth tables are a suitable way of writing this.

### Examples from algebra

On the set we consider two operations, addition and multiplication . These correspond to the following two link tables: ${\ displaystyle A = \ {0,1,2,3,4,5,6,7,8,9,10,11 \}}$${\ displaystyle a + b \, {\ bmod {5}}}$${\ displaystyle a \ cdot b \, {\ bmod {5}}}$

${\ displaystyle +}$ 0 1 2 3 4th
0 0 1 2 3 4th
1 1 2 3 4th 0
2 2 3 4th 0 1
3 3 4th 0 1 2
4th 4th 0 1 2 3
${\ displaystyle \ cdot}$ 0 1 2 3 4th
0 0 0 0 0 0
1 0 1 2 3 4th
2 0 2 4th 1 3
3 0 3 1 4th 2
4th 0 4th 3 2 1

Some properties of an inner two-digit link can easily be read from the link table: ${\ displaystyle \ circ \ colon A \ times A \ to A}$

Commutativity
The link is exactly then commutative , that is fulfilled for all , if the truth table symmetric about the main diagonal is. This is the case in both of the above examples.${\ displaystyle \ circ}$${\ displaystyle a \ circ b = b \ circ a}$${\ displaystyle a, b \ in A}$
Neutral element
One element is exactly then left neutral , that is fulfilled for all when the -line is a copy of the header. The same applies to a right-neutral element and the column. In the above example there is a neutral element on both sides. In the example there is an element that is neutral on both sides.${\ displaystyle e \ in A}$${\ displaystyle e \ circ a = a}$${\ displaystyle a \ in A}$${\ displaystyle e}$${\ displaystyle e}$${\ displaystyle e}$${\ displaystyle (A, +)}$${\ displaystyle 0}$${\ displaystyle (A, \ cdot)}$${\ displaystyle 1}$
Inverse elements
We assume from the previous example that there is a mutually neutral element for the link . At a given element is exactly then right inverse if true. The existence of such a right inverse can be seen from the fact that the element appears in the -line . The same applies to a left inverse and the column. In the example above , both sides are inversely closed . In the example has no inverse, every other element has exactly one inverse.${\ displaystyle e}$${\ displaystyle \ circ}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle a \ circ b = e}$${\ displaystyle a}$${\ displaystyle e}$${\ displaystyle a}$${\ displaystyle (A, +)}$${\ displaystyle 3}$${\ displaystyle 2}$${\ displaystyle (A, \ cdot)}$${\ displaystyle 0}$
Associativity
The link is associative if applies to all . Whether or not a link has this property is not immediately apparent when looking at your board and can only be checked by laborious trial and error.${\ displaystyle \ circ}$${\ displaystyle a \ circ (b \ circ c) = (a \ circ b) \ circ c}$${\ displaystyle a, b, c \ in A}$
Quasi-groups and Latin squares
A quasi-group is a non-empty set with a link , so that for all and in the equations and each have exactly one solution in . This is expressed in the link table by the fact that each row is a permutation of the header row and each column is a permutation of the input column. Such a table is also called a Latin square .${\ displaystyle Q}$${\ displaystyle \ circ \ colon Q \ times Q \ to Q}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle Q}$${\ displaystyle a \ circ x = b}$${\ displaystyle y \ circ a = b}$${\ displaystyle Q}$

For more examples of linking tables see: Klein's group of four , quaternion group , Sedenion , S3 (group) , A4 (group) .

## history

Linking tables were first used in group theory by Arthur Cayley . In a work of 1854 he calls it simple panels (engl. Tables ) and uses them for explaining groups. In his honor, linking tables are also called Cayley tables in group theory . For the construction of groups, however, linking tables are only suitable for very small groups, since systematic testing is hopelessly inefficient with a large number of elements. This approach was therefore supplemented and finally replaced in group theory by more powerful constructions, and no longer plays a role in theory today. The linking table of a group, however, leads directly to Cayley's theorem and thus to a natural starting point of the representation theory of groups.

• Arthur Cayley : "On the theory of groups, as depending on the symbolic equation θ n = 1", Philosophical Magazine , Vol. 7, pp. 40-47. Available online at GoogleBooks as part of his Collected Works .
• Arthur Cayley: On the Theory of Groups . In: American Journal of Mathematics , Vol. 11, No. 2 (January 1889), pp. 139–157, freely available online at JSTOR 2369415 .