Commutative aperiodic semigroups as multiset systems

The classification of finite commutative groups is relatively straightforward and has essentially already been achieved in terms of certain prime factorizations (which are of course multisets themselves). The harder problem then is to consider commutative aperiodic semigroups, so most of the work has went into examining them in detail. A brief description of some of these commutative aperiodic semigroups is therefore described below. All commutative associative operations are operations on multisets. Semilattices on the other hand are like operations on sets, and their characteristic operation is the closure of certain sets in a set system.

In order to generalize that to commutative aperiodic semigroups we need to define semigroups that can be described by their closure operations. To do this first we determine the idempotents by the non-repeating elements in the multisets of the set system, for these the product is equal to themselves. Otherwise the closure operation is the typical closure operation in the distributive lattice of multisets corresponding to the smallest parent. In this way, we can generalize the notion of a semilattice. There is one exceptional case size four, and then there are eighteen more of size five, and the exceptions increase within the class of aperiodic semigroups.

Size zero:

The only semigroup on zero elements corresponds to the empty set system {}.

Size one:

The only semigroup on one element is the trivial semigroup {{}}.

Size two:

There are two size two commutative aperiodic semigroups which are associated with the two types of multiset system: a set system and a multiset system. The monogenic aperiodic semigroup is clearly the multiset system because it is not idempotent and therefore it is not a semilattice.

• The semilattice: {{} {0}}
• The monogenic aperiodic semigroup : {{0} {0 0}}

Size three:

There are seven commutative aperiodic semigroups that have three elements. Since there is so few of these, it is still possible for us to give each of them different names. Since the tree semilattice is not unital, it does not include the identity element. The order of the multiset system is essentially the number of generators used to represent it.

• The total order semilattice: {{} {0} {0 1}}
• The tree semilattice: {{0} {1} {0 1}}
• The monogenic monoid: {{} {0} {0 0}}
• The monogenic semigroup plus zero: {{0} {0 0} {0 0 1}}
• The pseudozero semigroup : {{0} {1} {0 0 1}}
• The zero semigroup: {{0} {1} {0 1}}
• The monogenic semigroup: {{0} {0 0} {0 0 0}}

Size four:

Semilattices:

total order

[ [ 1, 1, 1, 1 ], 
  [ 1, 2, 2, 2 ], 
  [ 1, 2, 3, 3 ], 
  [ 1, 2, 3, 4 ] ]
{{} {0} {0 1} {0 1 2}}

weak order [1 2 1]

[ [ 1, 1, 1, 1 ], 
  [ 1, 2, 1, 2 ], 
  [ 1, 1, 3, 3 ], 
  [ 1, 2, 3, 4 ] ]
{{} {0} {1} {0 1}}

weak order [2 1 1]

[ [ 1, 1, 1, 1 ], 
  [ 1, 2, 2, 2 ], 
  [ 1, 2, 3, 2 ], 
  [ 1, 2, 2, 4 ] ]
{{0} {1} {0 1} {0 1 2}}

weak order [3 1]

[ [ 1, 1, 1, 1 ], 
  [ 1, 2, 1, 1 ], 
  [ 1, 1, 3, 1 ], 
  [ 1, 1, 1, 4 ] ]
{{0} {1} {2} {0 1 2}}

tree semilattice

[ [ 1, 1, 1, 1 ], 
  [ 1, 2, 1, 1 ], 
  [ 1, 1, 3, 3 ], 
  [ 1, 1, 3, 4 ] ]
{{0} {1} {1 2} {0 1 2}}

Near semilattices
Monogenic monoid + zero

[ [ 1, 1, 1, 4 ], 
  [ 1, 1, 2, 4 ], 
  [ 1, 2, 3, 4 ], 
  [ 4, 4, 4, 4 ] ]
{{} {2} {2 2} {2 2 4}}

Bi-idempotent near-zero semigroup + zero

[ [ 1, 1, 1, 4 ], 
  [ 1, 1, 1, 4 ], 
  [ 1, 1, 3, 4 ], 
  [ 4, 4, 4, 4 ] ]
{{2} {2 2} {2 2 3} {2 2 3 4}}

Monogenic semigroup + 0 + 0

[ [ 1, 1, 3, 4 ], 
  [ 1, 1, 3, 4 ], 
  [ 3, 3, 3, 3 ], 
  [ 4, 4, 3, 4 ] ]
{{2} {2 2} {2 2 4} {2 2 3 4}}

2-monogenic and 1-monogenic semigroups linked by a new zero element

[ [ 1, 1, 3, 3 ], 
  [ 1, 1, 3, 3 ], 
  [ 3, 3, 3, 3 ], 
  [ 3, 3, 3, 4 ] ]
{{4} {2} {2 2} {2 2 4 3}}

Monogenic monoid + identity

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 2, 2 ], 
  [ 1, 2, 3, 3 ],
  [ 1, 2, 3, 4 ] ]
{{} {3} {2 3} {2 2 3}}

Bi-idempotent near-zero semigroup + identity

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 2 ],
  [ 1, 1, 3, 3 ],
  [ 1, 2, 3, 4 ] ]
{{} {3} {2} {2 2 3}}

Exceptional tri-idempotent semigroup

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 2 ], 
  [ 1, 1, 3, 1 ], 
  [ 1, 2, 1, 4 ] ]
{{3} {4} {2 4} {2 2 3 4}}

Blockwise near-zero semigroup with total order

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 3, 3 ], 
  [ 1, 1, 3, 4 ] ]
{{4} {3 4} {2} {2 2 3 4}} 

Tri-idempotent near-zero semigroup

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 3, 1 ], 
  [ 1, 1, 1, 4 ] ]
{{2} {3} {4} {2 2 3 4}}

Max index two semigroups:
Zero semigroup + new zero (4 4 4 4)

[ [ 1, 1, 1, 4 ], 
  [ 1, 1, 1, 4 ], 
  [ 1, 1, 1, 4 ], 
  [ 4, 4, 4, 4 ] ]
{{2} {3} {2 2 3 3} {2 2 3 3 4}

Zero semigroup + identity (1 2 3 4)

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

{{} {2} {3} {2 2 3 3}}
Zero-semigroup plus a new idempotent element (1 2 2 4)

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 2 ],
  [ 1, 1, 1, 2 ], 
  [ 1, 2, 2, 4 ] ]
{{3} {4} {3 4} {3 3 4}}

Zero-semigroup plus a new idempotent element (1 1 3 4)

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 3 ], 
  [ 1, 1, 3, 4 ] ]
{{3} {3 4} {2} {2 2 3 3 4}}

Bi-idempotent near-zero semigroup (1 1 1 4)

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 4 ] ]
{{4} {2} {3} {2 2 3 3 4}}

Non-unique iteration

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 2, 2 ], 
  [ 1, 2, 3, 3 ], 
  [ 1, 2, 3, 3 ] ]
{{4} {4 4} {2 4 4} {2 2 4 4}}

One element nilpotent the other is notand otherwise the elements go to zero

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 3, 3 ], 
  [ 1, 1, 3, 3 ] ]
{{4} {4 4} {2} {2 2 4 4}}

Semigroups of max index three
Monogenic semigroup of index 3 with identity element

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 2 ], 
  [ 1, 1, 2, 3 ], 
  [ 1, 2, 3, 4 ] ]
{{} {0} {0 0} {0 0 0}}

Monogenic semigroup of index 3 with a new zero element

[ [ 1, 1, 1, 4 ], 
  [ 1, 1, 1, 4 ], 
  [ 1, 1, 2, 4 ], 
  [ 4, 4, 4, 4 ] ]
{{3} {3 3} {3 3 3} {3 3 3 4}}

Monogenic semigroup of index 3 with lesser idempotent

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 2, 1 ], 
  [ 1, 1, 1, 4 ] ]
{{4} {3} {3 3} {3 3 3 4}}

Monogenic with semigroup with element that is index two but for which 3,4 is not directly nilpotent

 
[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 2 ], 
  [ 1, 1, 2, 2 ] ]
{{4} {3} {3 4 4} {3 3 4 4 4}}

Monogenic semigroup plus a zero

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 2 ] ]
{{4} {3} {4 4} {3 3 4 4 4}}

Monogenic semigroup with lesser element that is nilpotent but for which the generator and the new element do not directly go to zero Reducibles: 1,2

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 2, 2 ], 
  [ 1, 1, 2, 2 ] ]
{{3} {4} {3 3 4 4} {3 3 3 4 4 4}}

The aperiodic monogenic semigroup:
Aperiodic monogenic semigroup:

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 3 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 3, 1, 2 ] ]
{{0} {0 0} {0 0 0} {0 0 0}}