Distinctions among limit points

Firstly, I would like to briefly consider the different types of limit points and the nature of their differences. A topological space can be defined entirely in terms of its limit points, as the limit points are precisely the points generated by the closure of a set. In a way topological spaces exist to describe limit points. In typical applications, limits are defined in terms of the behavior of sequences as they approach infinity, so these limits are generated by an infinite process. The definition of a topological space, however, allows for limits that are generated by individual elements.

• Special limit points: limit points that are generated by a finite set
• Analytic limit points: limit points that are generated by an infinite set

The special limit points are generated entirely be the specialization preorder of the topology. This is really then about the relationship between order, topology, and the infinite. Topology only gains its own identity as distinguished from order theory when dealing with the limits of infinite sets. Alexandrov topologies are entirely generated by their special limit points and they are determined by their specialization preorder. Analytic limits arise in the purest sense in order topology when dealing with suborders of non-hereditary-discrete total orders (total orders other then $\mathbb{Z}$). These topologies are Frechet because they are generated entirely by analytic limit points, and metric topologies share this property. Hopefully, this clarifies the nature of topology.

Max order one multiset systems

In a previous post I described how the ordering of distinct max order one multiset systems is a disjoint union of total orders. This describes the order type of the support of a max order one multiset system but it does not completely describe the multiset system type of the max order one multiset system.

Therefore, we need to further consider max order one multiset systems, and the multisets of prime powers, which often emerge from commutative groups. Consider as an example {2,2,4,4,3,3,27} then in this case we can partition this by the support of each of these multisets and therefore we will get {2,2,4,4} and {3,3,27}. Then each of these has their own signature defined by the multiset of exponents of each of these multisets of prime factorizations in this case, though the multiplicative concept of a positive integer and a finitary multiset are equivalent. The multiset system type of these is then the multiset of exponent signatures of these components.

So for the example of {2,2,4,4} we will have {1,1,2,2} and for {3,3,27} we will have {1,1,3}. The overall type is then {{1,1,2,2},{1,1,3}. This perhaps demonstrates that multisets of signatures will play a key role in understanding multiset systems. The only other detail is the multiplicity of the empty set, in the not necessarily nullfree case, which is simply a non-negative integer. These multisets of signatures can also be acquired from the membership signatures of each dimember of a multiset system, which generalizes signatures of set systems themselves.

A larger example is {2,2,4,3,3,9,5,125,125,3125,3125,3125,7,343} which produces the multiset system type {{1,1,2},{1,1,2},{1,3,5},{1,3}}. This demonstrates that this multiset of signatures can have repetition as we can see that {2,2,4} is isomorphic to {3,3,9} as a multiset system of prime factorizations. Together this fully defines these multiset systems that emerge from commutative groups. The height of each element of each total order is the support size of each of these multisets, so we can get the order type as well from this full description.

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}}