[loops] 4 Basic methods and attributes

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# 4 Basic methods and attributes

### Sections

1. About Cayley tables
2. Testing Cayley tables
3. Canonical and normalized Cayley tables
4. Creating quasigroups and loops manually
5. Creating quasigroups and loops from a file
6. Conversions
7. Products of loops
8. Opposite quasigroups and loops
9. Basic attributes
10. Basic arithmetic operations
11. Powers and inverses
12. Associators and commutators
13. Generators

We describe the basic core methods of the LOOPS package in this chapter. The methods discussed here (and more) are declared and implemented in files `quasigrp.gd` and `quasigrp.gi`, respectively.

## 4.1 About Cayley tables

Let X={x1,...,xn} be a set and · a binary operation on X. Then an n by n array with rows and columns bordered by x1, ..., xn, in this order, is a Cayley table , or a multiplication table of ·, if the entry in the row xi and column xj is xi·xj.

A Cayley table is a quasigroup table if it is a Latin square , i.e., if every entry xi appears in every column and every row exactly once.

An annoying feature of quasigroup tables in practice is that they are often not bordered, and it is up to the reader to figure out what is meant. Throughout this manual and in LOOPS, we therefore make the following assumption: All distinct entries in a quasigroup table must be integers, say x1 < x2 < … < xn, and if no border is specified, we assume that the table is bordered by x1, ..., xn, in this order. Note that we do not assume that the distinct entries x1, ..., xn form the interval 1, ..., n. The significance of this observation will become clear in Chapter Some methods based on permutation groups.

Finally, we say that a quasigroup table is a loop table if the first row and the first column are the same, and if the entries in the first row are ordered in an ascending fashion.

## 4.2 Testing Cayley tables

A square array with integral entries is called a matrix in GAP. The following synonymous operations test if a matrix T is a quasigroup table, as defined above:

- `IsQuasigroupTable(` T`) O`
- `IsQuasigroupCayleyTable(` T`) O`

  The following synonymous operations test if a matrix T is a loop table:

- `IsLoopTable(` T`) O`
- `IsLoopCayleyTable(` T`) O`

  We would like to call attention to the fact that the package GUAVA also has some operations dealing with Latin squares. In particular, `IsLatinSquare` is declared in GUAVA.

  ## 4.3 Canonical and normalized Cayley tables

  Although we do not assume that a quasigroup table with distinct entries x1 < … < xn satisfies xi=i, it is often desirable to present quasigroup tables in the latter way. The rather general operation

- `CanonicalCayleyTable(` T`) O`

  takes any Cayley table T with distinct entries x1 < … < xm, and returns a Cayley table in which xi has been replaced by i.

  The operation

- `NormalizedQuasigroupTable(` T`)`

  makes a quasigroup table T into a loop table by:

  - first calling `CanonicalCayleyTable` to rename the entries to 1, ..., n,
  - then permuting the columns of T so that the first row reads 1, ..., n,
  - and then permuting the rows of T so that the first column reads 1, ..., n.

  ## 4.4 Creating quasigroups and loops manually

  When T is a quasigroup table, the corresponding quasigroup is obtained by

- `QuasigroupByCayleyTable(` T`) O`

  Since `CanonicalCayleyTable` is called within the above operation, the resulting quasigroup will have a Cayley table with distinct entries 1, ..., n.

  Here is the analogous operation for a loop table T:

- `LoopByCayleyTable(` T`) O`

  ## 4.5 Creating quasigroups and loops from a file

  Typing a large multiplication table manually is tedious and error-prone. We have therefore included a universal algorithm in LOOPS that reads multiplication tables of quasigroups from a file.

  Instead of writing a separate algorithm for each common format, our algorithm relies on the user to provide a bit of information about the input file. Here is an outline of the algorithm, with file named F and a string D as arguments on the input:

  - read the entire content of F into a string S,
  - replace all end-of-line characters in S by spaces,
  - replace by spaces all characters of S that appear in D,
  - split S into maximal substrings without spaces, called chunks,
  - recognize distinct chunks (let n be the number of distinct chunks),
  - if the number of chunks is not n2, report error,
  - construct the multiplication table by assigning numerical values 1, ..., n to chunks, depending on their position among distinct chunks.

  The following examples clarify the algorithm and document its versatility. All examples are of the form F+D⇒ T, meaning that an input file containing F together with the string D produce multiplication table T.

  Example 1: Data does not have to be arranged into an array of any kind.

  0
  1
  2
  1
  2
  0
  2
  0
  1
  +
  ""
  ⇒
  1
  2
  3
  2
  3
  1
  3
  1
  2

  Example 2: Chunks can be any strings.

  red
  green
  green
  red
  +
  ""
  ⇒
  1
  2
  2
  1

  Example 3: A typical table produced by GAP is easily parsed by deleting brackets and commas.

  [ [0, 1], [1, 0] ]
  +
  "[,]"
  ⇒ |
  1
  2
  2
  1

  Example 4: A typical TeX table with rows separated by lines is also easily converted. Note that we have to use \\ to make sure that every occurrence of \ is deleted, since \\ represents the character \ in GAP.

  x& y& z\cr
  y& z& x\cr
  z& x& y
  | +
  "\\cr&"
  ⇒
  1
  2
  3
  2
  3
  1
  3
  1
  2

  And here are the needed \LOOPS commands:

- `QuasigroupFromFile(` F`,` D`) O`
- `LoopFromFile(` F`,` D`) O`

  ## 4.6 Conversions

  As we have already briefly mentioned, we provide operations that convert between magmas, quasigroups, loops and groups, provided such conversions are possible.

  If M is a declared magma that happens to be a quasigroup, the corresponding quasigroup is returned via

- `AsQuasigroup(` M`) O`

  If M is a magma that happens to be a quasigroup, the operation

- `AsLoop(` M`) O`

  returns a loop L as follows:

  - if M possesses a neutral element e and f is the first element of M, then L is an isomorphic copy of M via the transposition (e,f),
  - if M does not posses a neutral element, L is returned as `PrincipalLoopIsotope(` M`,` M.1`,` M.1`)`.
  Here,

- `PrincipalLoopIsotope(` Q`,` f`,` g`) O`

  is the principal isotope of a quasigroup Q using elements f, g of Q, as explained in Section Homomorphisms and homotopisms.

  Of course, one can obtain a loop from M in different ways, for instance by normalizing the Cayley table of M. The three approaches mentioned here can result in different loops in general.

  When M is a declared magma that happens to be a group, then the corresponding group is returned by

- `AsGroup(` M`) A`

  Note that the conversions work in both directions, not just toward more special structures. Thus, if G is a declared group, then `AsLoop(` G`)` returns the corresponding loop, for instance.

  ## 4.7 Products of loops

  noindentLet L1, ..., Ln be a list consisting of loops and groups, where n ≥ 1. Then

- `DirectProduct(` L1`, ...,` Ln`) O`

  returns the direct product of L1, ..., Ln.

  If there are only groups among L1, ..., Ln, a group is returned. Otherwise a loop is returned. If n=1, L1 is returned.

  ## 4.8 Opposite quasigroups and loops

  When Q is a quasigroup with multiplication ·, the opposite quasigroup of Q is a quasigroup with the same underlying set as Q and with multiplication * defined by x*y=y·x.

  Since the quasigroup-theoretical concepts are often chiral (cf. left Bol loops versus right Bol loops), it is useful to have access to the opposite quasigroup of Q:

- `Opposite(` Q`) O`

  ## 4.9 Basic attributes

  We associate many attributes with quasigroups in order to speed up computation. This section lists the basic attributes of quasigroups and loops.

  The list of elements of a quasigroup Q is obtained by the usual command

- `Elements(` Q`) A`

  The Cayley table of a quasigroup Q is returned with

- `CayleyTable(` Q`) A`

  One can use `Display( CayleyTable(` Q`) )` for pretty matrix-style output of small Cayley tables.

  The neutral element of a loop L is obtained via

- `One(` L`) A`

  If you want to know if a quasigroup Q has a neutral element, you can find out with the standard function for magmas

- `MultiplicativeNeutralElement(` Q`) A`

  The size of a quasigroup Q is calculated by

- `Size(` Q`) A`

  When L is a power-associative loop , i.e., the orders of elements are well-defined in L, the exponent of L is the smallest positive integer divisible by orders of all elements of L. The following attribute calculates the exponent without testing for power-associativity:

- `Exponent(` L`) A`

  Here is an example for operations and attributes mentioned sofar:

      gap> A := [ [ 1, 2 ], [ 3, 4 ] ];; B := [ [ 8, 5 ], [ 5, 8 ] ];;
      gap> [ IsQuasigroupTable( A ), IsQuasigroupTable( B ), IsLoopTable( B ) ];
      [ false, true, false ]
      gap> Q := QuasigroupByCayleyTable( B );; CayleyTable( Q );
      [ [ 2, 1 ], [ 1, 2 ] ]
      gap> CanonicalCayleyTable( B );
      [ [ 2, 1 ], [ 1, 2 ] ]
      gap> NormalizedQuasigroupTable( B );
      [ [ 1, 2 ], [ 2, 1 ] ]
      gap> LoopByCayleyTable( last );
      <loop of order 2>
      gap> [ IsQuasigroup( Q ), IsLoop( Q ), Size( Q ), Elements( Q )]
      [ true, false, 2, [ q1, q2 ] ]
      gap> IsQuasigroupElement( Elements( Q )[ 2 ] );
      true
      gap> AsLoop( Q );
      <loop of order 2>

  ## 4.10 Basic arithmetic operations

  Each quasigroup element in GAP knows which quasigroup it belongs to. It is therefore possible to perform arithmetic operations with quasigroup elements without referring to the quasigroup. All elements involved in the calculation must belong to the same quasigroup.

  Two elements x, y of the same quasigroup are multiplied by x*y in GAP. Since multiplication of elements is ambiguous in the nonassociative case, we always multiply elements from left to right, i.e., x*y*z means ((x*y)*z). Of course, one can specify association by parentheses.

  Universal algebraists introduce two additional operations for quasigroups. Namely the left division x\y satisfying x·(x\ y)=y, and the right division x/y satisfying (x/y)·y=x. These two operations can be found in LOOPS as:

- `LeftDivision(` x`,` y`) O`
- `RightDivision(` x`,` y`) O`

  When Q is a quasigroup, x is an element of Q, and S is a list of elements of Q, then

- `LeftDivision(` S`,` x`) O`
- `LeftDivision(` x`,` S`) O`
- `RightDivision(` S`,` x`) O`
- `RightDivision(` x`,` S`) O`

  returns the list of elements obtained by performing the respective division of S by x, or of x by S, using one element of S at a time.

  We also support / in place of `RightDivision`. But we do not support \ in place of `LeftDivision`.

  To obtain the Cayley tables for the operation left division or right division, use

- `LeftDivisionCayleyTable(` Q`) O`
- `RightDivisionCayleyTable(` Q`) O`

  ## 4.11 Powers and inverses

  Powers of elements are not well-defined in quasigroups, since bracketing can matter even for a single element. We say that the quasigroup Q is power-associative , if for any x ∈ Q, the submagma generated by x is associative.

  For magmas and positive integer exponents, GAP defines the powers in the following way: x1=x, x2k=(xk)·(xk) and x2k+1=(x2k)· x. One can easily see that this returns xk in log2(k) steps. For LOOPS, we have decided to keep this method, hoping that everybody will use it with care for non power-associative quasigroups.

  Let x be an element of a loop L with neutral element 1. Then the left inverse xλ of x is the unique element of L satisfying xλ x=1. Similarly, the right inverse xρ satisfies xxρ=1. If xλ=xρ, we call x−1=xλ=xρ the inverse of x.

- `LeftInverse(` x`) O`
- `RightInverse(` x`) O`
- `Inverse(` x`) O`

  The following examples illustrates the usage of arithmetic operations. `MoufangLoop` will be explained in Chapter Libraries of small loops. In this example, `M.i` coincides with `Elements( M )[ i ]`.

      gap> M := MoufangLoop( 12, 1 );; x := M.2;
      l2
      gap> [ x * M.3, x^2, x^(-1), Inverse( x ) ];
      [ l4, l1, l2, l2 ]
      gap> One( M ) = LeftDivision( x, x );
      true

  ## 4.12 Associators and commutators

  Let Q be a quasigroup and x, y, z ∈ Q. Then the associator of x, y, z is the unique element u such that (xy)z=(x(yz))u. The commutator of x, y is the unique element v such that xy=(yx)v.

- `Associator(` x`,` y`,` z`) O`
- `Commutator(` x`,` y`) O`

  ## 4.13 Generators

  The following two attributes are synonyms of `GeneratorsOfMagma`:

- `GeneratorsOfQuasigroup(` Q`) A`
- `GeneratorsOfLoop(` L`) A`

  As usual in GAP, one can refer to the ith generator of a quasigroup Q by `Q.i`. Note that it is not necessarily the case that `Q.i = Elements( Q )[ i ]`, since the set of generators can be a proper subset of the elements.

  It is easy to prove that a quasigroup of order n can be generated by a subset containing at most log2n elements. When Q is a quasigroup

- `GeneratorsSmallest(` Q`) A`

  returns a generating set {q0, ..., qm} of Q such that Q0=∅, Qm=Q, Qi=〈q1, ..., qi 〉, qi+1 is the largest element of Q\Qi.

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