2019 year in review

The most important point of consideration during this past year was the theory of commutative operations. Commutative operations can be defined as functions of unordered pairs rather then as functions of ordered pairs. This requires the notion of a multiset which can be defined as an unordered collection of elements. Sets have the limitation that they exclude repetition, but after the generalization of multisets they can be included amongst multisets as part of an ontology of unordered collections.

The theory of commutative semigroups emerges from the study of commutative operations. One of the first things one notices is that semilattices are included among the commutative semigroups. This is how we can then get to the idea of commutative operations that generalize semilattices. These need to be aperiodic (group free) semigroups. These operations can be defined as functions that produce an upper bound of elements of a partial order that need not be the least upper bound, because upper bounding producing functions are aperiodic.

Many commutative operations of an aperiodic type actually arise directly out of semilattices from the properties of elements (such as from monotone functions). This occurs in the complement of the partition lattice for example. Partial orders are known to strictly order things from minimal to maximal, so the operations that correspond to them are aperiodic. That is to say, aperiodic operations correspond to partial orders. There are of course also commutative groups such as the cyclic group. The theory of finite commutative groups is of course well known.

Multisets can be defined as elements of the free commutative semigroup on a universal generating set. Multisets can therefore be concatenated counting repetition using this semigroup. Using the concept of multisets we can consider the nature of commutative aperiodic semigroups (especially small commutative aperiodic semigroups) using multiset systems. By analogy with the representation of semilattices as closure operations on set systems we can define certain commutative aperiodic semigroups as closure operations on multiset systems subject to certain index limitations. Of course, in the free case no closure operation is necessary and we have an operation fully defined from multiset combination.

The important concept of degree reductions produces a partial order on set systems and their generalizations based upon the idea of taking subsets of members rather then just subsets of the set. It is noticeable that the combinational multiset of combinatorial set systems is a monotone function of these degree reductions, so we can consider certain set systems as partitions of a multiset. For example, the kuratowski pair is the unique set theoretic partition of the kuratowski pair multiset. We don't even need set systems then, and we can define ordered pairs directly from multisets.

Finite multisets of course have signatures associated with them. So for example, we can see that kuratowski pair multisets have certain special signatures. These signatures can be ordered by the Young's lattice so that they are a monotone function on multisets in the same way that cardinal numbers are a monotone function on finite sets. Young's lattice therefore plays an important role in multiset theory.

There are two main ways to combine two commutative operations together (1) into a distributive lattice and (2) into a commutative ring. These commutative operations are combined by the distributive law, which ensures that polynomials do not need to be nested (hence the importance of polynomials in commutative and boolean algebra). Distributive lattices are essential in set theory and multiset theory. Both multisets and their signatures form distributive lattices. Elements of distributive lattices can be represented as sets, for example this naturally leads to the definition of von neumann ordinals. Commutative rings are used in the definition of arithmetic and they have important properties related to factorizations (which are actually multisets).

Natural arithmetic $(\mathbb{N},+,*)$ is not a commutative ring because it lacks additive inverses, but it is a commutative semiring. Both natural addition and natural multiplication are (with the single exception of multiplication by zero) free commutative operations defined by combining multisets, so the extent to which arithmetic is multiset theory is considerable. This is ensured by the fundamental theorem of arithmetic which states that every positive integer is uniquely factored into a multiset of prime numbers. Extra details are of course added on by the different types of commutative operations and inverses. But algebraic number theory is still essentially commutative.

I covered too many things in the past year to cover them all here effectively, but I mainly want to draw attention to multisets, multiset systems, commutative semigroups, commutative rings, distributive lattices, and related concepts. The main defining feature of these structurse is that like sets they lack an ordering on their elements.

Torsion free arithmetic

In the previous post, we noted that natural arithmetic $(\mathbb{N},+,*)$ is aperiodic in both addition and multiplication. As aperiodic operations they are of course directly related to partial ordering relations. The addition operation is directly related to the standard total ordering and the multiplication operation is directly related to the divisibility partial ordering with zero as the maximal element. Both of these are essentially distributive lattices as partial orders, which are also multiset inclusion lattices. The operations are multiset combination because natural addition and multiplication are both free. That is the situation with natural arithmetic. In the case of the integers $(\mathbb{Z},+,*)$ neither addition or multiplication is aperiodic, so we have to consider the related property of being torsion-free.

Torsion-free addition: the addition operation on the integers $(\mathbb{Z},+)$ is torsion-free. The reason for this is that all the negative integers are simply mirror images (or algebraic inverses) of their positive sides and so they still have the same infinite torsion-free behavior. So this means that the origin of periodic arithmetic does not lie in addition.

Partially torsion multiplication: the multiplication operation on the integers $(\mathbb{Z},*)$ is essentially the same as the on the natural numbers except that there are two elements ${-1,1}$ which are in a product with the ordinary integers. Furthermore, these two elements ${-1,1}$ form a cyclic group $C_2$ which means that $-1$ alone has periodic properties among the integers and only with respect to multiplication.

Natural arithmetic is nonperiodic

An aperiodic semigroup is a semigroup that contains no non-trivial subgroups. This means that for all elements $x$ there is a point at which further iteration of the element produces no effect or in other words $x^{n+1}=x^n$. Elements of this form a clearly aperiodic, and that is already established in the literature. A separate issue, however, is rather or not elements that can be iterated infinitely are considered periodic. It is clear that since these elements continue to infinity without any repetition they cannot be considered periodic either. As a result, a separate concept is nonperiodic commutative semigroups.

We can immediately see that $(\mathbb{N},+)$ and $(\mathbb{N},*)$ are nonperiodic. In $(\mathbb{N},+)$ the first element 0 is idempotent, and the remaining elements continue infinitely. In $(\mathbb{N},*)$ the first elements are 0 and 1 both of which are idempotent and then the rest continue infinitely. The only difference is that multiplication has more idempotents. As a result, natural arithmetic has no unorderly aperiodic behavior.

Non-periodic semigroups have the most order-theoretic behavior among the class of semigroups. Non-periodic semigroups can be defined by the composition of extensive monotone functions of a partial order. The partial order of addition is the natural partial order, and the partial order of multiplication is the divisibility partial order. The divisibility partial order is a suborder of the natural ordering except that zero is considered maximal. With respect to these orderings, we can see that addition strictly increases the natural ordering and multiplication strictly increases the divisibility ordering. Multiplication by zero simply transforms an element to the maximal element in the partial order.

It is also noticeable that the addition and multiplication semigroups are therefore related to semilattices. In particular, the maximum semilattice and the least common multiple semilattice. Considering these as upper bounds we see that $max(a,b) <= (a+b)$ and $lcm(a,b)| (a*b)$ as maximum is the least upper bound of its ordering and addition is a much greater bound and likewise lcm is the least upper bound of divisibility and multiplication is an upper bound that is non-minimal.

The nonperiodic and orderly behavior of the arithmetic of the natural numbers is the basis of the connection between arithmetic and logic. Both the natural arithmetic operations and the lattice operations are commutative, associative, and nonperiodic. This is why when we have two disjoint sets $A$ and $B$ and we take their union the cardinality is equal to the sum of the cardinality of the two of them. Addition is an abstraction of the operation of joining two sets in set theory and classical logic. It abstracts away the elements of a set and it tells us about their cardinalities. In the same way, multiplication is an abstraction of the joining of partitions in the co-partition lattice defined in partition logic. In this sense, arithmetic exists to benefit the understanding of logic.

Commutative aperiodic semigroups theory

The idea of commutative aperiodic semigroups theory emerged from considerations of generalizations of semilattices. The first aspect of this is that the idempotent property must be sacrificed in order to consider various generalizations that allow for repetition. Towards that end, we first consider commutative aperiodic semigroups which are distinct join preserving, which therefore are essentially equivalent to semilattices except for the condition that iteration and repetition is allowed. This is the general case until the commutative aperiodic semigroup of order 4, which is ordered by the weak order [2 1 1] appears.

Well considering this, another direction of thought came to me based upon the idea of bounds. The idea of a join and a meet are defined by the least upper bound or the greatest lower bound of two elements. But what happens if you relax the least or greatest condition? Then in that case you get a commutative aperiodic semigroup, so we can see how commutative aperiodic semigroups and perhaps even eventually other types of commutative semigroups can play a role in order theory.

In the place of semilattices we can instead produce a partially ordered set of commutative aperiodic semigroups on a partial order, determined by the leastness of the upper bound produced by the semigroup. This leads to the notion of a greatest upper bound, naturally this would seem trivial, but actually it isn't if we restrict ourselves to the condition that the result that the partial order is preserved by the algebraic preorder of the semigroup. This leads to our understanding of commutative aperiodic semigroups.

The trivial case: in the trivial case we know that there is only one possible semigroup so distinctions don't matter

The total order T2: there is only two cases the semilattice and the non-semilattice

The total order T3: in this case the commutative aperiodic semigroups with the total order on three elements can actually been ordered in a four-element diamond shape. The most semilattice like is the semilattice itself, then there are too cases that relax the semilattice property somewhat by making either the minimal element or the middle element a generator of its parent. The least semilattice like is the monogenic semigroup on three elements. The monogenic semigroup is essentially the greatest upper bound as compared to the least upper bound of the semilattice.

The tree order [2 1]: in this case the three types of commutative aperiodic semigroup can be determined by the number of idempotent elements they have. The more idempotent elements the more semilattice like the semigroup is. There are three cases the semilattice, the case where a single element is nilpotent with index two, and the zero semigroup itself which is the least semilattice like and which has two elements of index two which generate the zero element. This produces a semilatticeness partial order on the types of semigroups that have this algebraic partial order.

The general principle proceeds accordingly for the larger commutative aperiodic semigroups. The exceptional semigroup on four elements is the least semilattice like semigroup on the partial order [2 1 1]. The property of not preserve distinct joins makes a semigroup even less semilattice like, so it can be considered to be part of the hierarchy of different properties related to the ordering of semigroups based upon their semilatticeness.

Remainders from roots

I described how multisets can be divided to get a quotient and a remainder. This generalizes the process of division of a number, which is actually equivalent to dividing a equal multiset. The process of dividing a multiset can be generalized to other multisets though, which produces a different process. The first immediate thought is that this can be applied to prime factorizations. So for example if we prime factorize 24 we get {2,2,2,3} and if we divide it 2 we get {2} with remainder {2,3}. This division of the prime factorization multiset is essentially the same as taking roots.

So if we take the square root of 24 we get 2 with a remainder of 6, and it is expressed as $2\sqrt{6}$. I particularly like the number 360 because of its unique minimal prime factorization {2,2,2,3,3,5} which forms a progression multiset. If we divide this by two we get $6$ with a remainder of 10 so it is $6\sqrt{10}$. In the case of a cube root we get $2$ with a remainder of $45$ so $2 \sqrt[3]{45}$. In any case, the remainder is the object still in the root symbol and the quotient is the part which is not.

Ordinary division is essentially additive division so when computing division the remainder is added to the quotient as a fraction and the introduction of fractions is what distinguishes the result from the ordinary integers. Roots are multiplicative division so the remainder is multiplied by the quotient, rather then added to it. The remainder is then the algebraic part that distinguishes it from the other rational part.

I have seen the remainder be used to refer to the fractional part of a root, so the square root of 24 would then be 4 with a remainder of 0.898979... going on infinitely in a non-periodic manner. It is useful to consider 4 as the square root of the smallest square number less then the number, but it is wrong to consider 0.898979... to be the remainder. Instead this is the fractional part of the root. This is largely an issue of terminology, but it is interesting nonetheless.

Multiset division and remainder

The operations of division and remainder, so familar to us because of their use with the natural numbers, can be generalized to multisets. Numerous familiar concepts can be generalized and better understood through the use of multisets, which are relatively underused in the current situation. The first thing one realizes is that we can use this to divide and get the remainder of the prime factorization multiset of a number. So for example with 8 we can divide its factorization by 2 to get 2, and a remainder of 2 which is actually how we compute roots anyways to get $2\sqrt{2}$. So this means that nth-roots are actually based upon dividing a multiset, though it was never taught to me that way and I don't think its that common to describe it like that. Here is some clojure code that deals with the general case of dividing multisets.

(defn multiset-division
  [coll divisor]

  (Multiset.
   (into
    {}
    (for [elem (support coll)
          :let [n (multiplicity coll elem)]
          :when (<= divisor n)]
      [elem (quot n divisor)]))))

(defn multiset-remainder
  [coll divisor]

  (Multiset.
   (into
    {}
    (for [elem (support coll)
          :let [n (multiplicity coll elem)]
          :when (not (zero? (mod n divisor)))]
      [elem (mod n divisor)]))))

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.