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Magma

  

Enumerate

  

enumerate small finite magmas up to isomorphism

 

Calling Sequence

Parameters

Options

Description

Examples

Compatibility

Calling Sequence

Enumerate( n )

Enumerate( n, options )

Parameters

n

-

an integer greater than 1

options

-

optional property specifications

Options

• 

Several options are provided to filter the kind of magmas enumerated.  For the most part, these circumscribe various equational identities that the enumerated magmas are required to satisfy.

  

The recognized options are as follows:

alternative

enumerate alternative magmas

anticommutative

enumerate anti-commutative magmas

associative

enumerate associative magmas

band

enumerate bands (idempotent semigroups)

commutative

enumerate commutative magmas

distributive

enumerate distributive magmas

group

enumerate groups

identity

enumerate magmas in which 1 is the identity element

idempotent

enumerate idempotent magmas

inverses

enumerate magmas with identity in which elements are invertible

kei

enumerate kei (involutary quandles)

leftbol

enumerate left Bol magmas

leftcancellative

enumerate left cancellative magmas

leftdistributive

enumerate left distributive magmas

leftinvolutary

enumerate left involutary magmas (X(XY) = Y)

leftsemimedial

enumerate left semimedial magmas

loop

enumerate loops

medial

enumerate medial magmas

monoid

enumerate monoids

quandle

enumerate quandles

quasigroup

enumerate quasigroups

rack

enumerate racks

rightbol

enumerate right Bol magmas

rightcancellative

enumerate right cancellative magmas

rightdistributive

enumerate right distributive magmas

rightinvolutary

enumerate right involutary magmas (X(YY) = X)

rightsemimedial

enumerate right semimedial magmas

semilattice

enumerate semilattices (commutative and idempotent semigroups)

squag

enumerate squags (idempotent Steiner magmas)

steiner

enumerate Steiner magmas

szasz

enumerate Szasz (uniquely nonassociative) magmas

zeropotent

enumerate zeropotent magmas

• 

The presence of multiple property options implies the conjunction of all the passed option properties.  Consequently, contradictory properties, such as associative and szasz will result in no magmas being enumerated.

• 

process = processor

  

The process = processor option allows you to specify a procedure to be called on each representative of the isomorphism classes of magmas enumerated. The procedure (processor) will be passed the Cayley table as the first argument and the order of the magma (a positive integer) as the second argument, and it can do whatever processing you want with the isomorphism class representative. Note that, if you want the processor argument to keep a copy of the Cayley table, then it must make an explicit copy (using, for instance, the copy command), since the same Array will be passed on each call, but with different numerical entries each time. The processor procedure should not attempt to modify the passed Array representing the Cayley table of the magma; changes will not be reflected in the enumeration.

• 

test = tester

  

The test = tester option allows you to specify a nonbuilt-in predicate to be used to prune the search tree. It is passed a partially completed Cayley table for a magma of the specified order, as well as the order.  Only nonzero elements of the Cayley table are to be tested, as zero elements are, at the time the supplied predicate is called, deemed to be undetermined. It will be called after all the built-in tests have been performed.  Therefore, it is safe to assume that partial Cayley tables passed to it already satisfy all partial checks implied by any other properties specified. The tester predicate you pass with this option will be called as a procedure with the partially completed Cayley table as the first argument, and the order of the magma (a positive integer) as the second argument.  The partially completed Cayley table has the form of a two-dimensional array with datatype equal to integer[4] and with C_order storage order.  Thus, it is possible to use a compiled procedure as the value of this option.

Description

• 

The Enumerate command enumerates small (finite) magmas, optionally subject to certain constraints, described below. By default, it simply counts the number of such structures, up to isomorphism.

  

When called with no options or optional arguments, the Enumerate command simply counts the number of isomorphism classes of magmas of order n.

Examples

withMagma:

Count the number of two-element magmas, up to isomorphism

Enumerate2

10

(1)

Count the number of commutative and associative four-element magmas, up to isomorphism

Enumerate4,'commutative','associative'

58

(2)

Build a list of the three-element commutative and associative magmas.

LEnumerate3,'associative','commutative','output'='list'

L:=111111111,111111112,111111113,111112123,111121113,111122122,111122123,111123132,113113331,113123331,122211211,123231312

(3)

andmapIsAssociative,L

true

(4)

andmapIsCommutative,L

true

(5)

Count, and print compactly, the groups of order 4.

Enumerate4,'group','process'=m,n→lprintconvertm,'listlist'

[[1, 2, 3, 4], [2, 1, 4, 3], [3, 4, 1, 2], [4, 3, 2, 1]]
[[1, 2, 3, 4], [2, 1, 4, 3], [3, 4, 2, 1], [4, 3, 1, 2]]

2

(6)

Count the Moufang loops of order 6.

LEnumerate6,leftbol,rightbol,loop

L:=2

(7)

Count connected quandles of order 5.

isconnected := proc( m, n )
   local i, gens;

   try
     gens := map( convert, {seq}( convert( m[ .., i ], 'list' ), i = 1 .. n ), 'disjcyc' )
   catch "not a permutation":
     return true
   end try;
   evalb( nops( group:-orbit( permgroup( n, gens ), 1 ) ) = n )
end proc:

Enumerate5,'quandle','test'=isconnected

3

(8)

Determine which groups of order 9 have the property that each member of the group is a square.  The passed procedures must ignore currently undefined entries (represented by zeros) in the table and return false just in the case that a duplicate diagonal entry is found.  Otherwise, true is returned to indicate to the enumeration engine that it should continue trying to complete the Cayley table.

Enumerate( 9, 'group', 'test' = proc( m, n )
 local i, L := remove( type, [seq]( m[ i, i ], i = 1 .. n ), 0 );
 evalb( nops( L ) = nops( {op}( L ) ) ) end proc );

2

(9)

Count the loops of order 6 such that every square commutes with every member of the loop.  This illustrates the use of a procedure with the option autocompile.

sqcomm := proc( m :: Array( datatype = integer[4], order = C_order ), n :: posint )
  option autocompile;
  local i, j, z;

  for i from 1 to n do
    z := m[ i, i ];
    if z <> 0 then # ignore undefined entries
      for j from 1 to n do
        if m[ z, j ] <> 0 and m[ j, z ] <> 0 then # ignore undefined entries
          if m[ z, j ] <> m[ j, z ] then
            return false
          end if
        end if
      end do
    end if
  end do;
  true
end proc:

LEnumerate6&comma;&apos;loop&apos;&comma;&apos;test&apos;&equals;&apos;sqcomm&apos;&comma;&apos;output&apos;&equals;&apos;list&apos;&colon;

nopsL

15

(10)

How many of these are not commutative?

nopsremoveIsCommutative&comma;L

7

(11)

Compatibility

• 

The Magma[Enumerate] command was introduced in Maple 15.

• 

For more information on Maple 15 changes, see Updates in Maple 15.

• 

The semilattice option was introduced in Maple 16.

• 

For more information on Maple 16 changes, see Updates in Maple 16.

See Also

AreIsomorphic

autocompile

IsAssociative

IsCommutative

Magma

orbit

permgroup

 


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