Sanctuary

## Introduction

A **Sanctuary** (or **Sanctum**) is fairly elementary algebraic structure that acts as the simplest (and historically oldest) part of a zeroid theory. Zeroid Theory began with the operation of multiplication and then later added addition. Multiplication is the natural domain of Zeroid Theory since its a theory about relative size and multiplication is about scaling and relative size. Addition on the other hand, is based around notions of relative position.

Addition took much longer to incorporate into Zeroid Theory than multiplication and it's implementation is less natural and straight forward. While addition leads to the richest and most interesting results, a core "inner sanctum" of zeroid theory that we can rely on is in the multiplication operation. A Sanctuary gives us this core structure. It is much simpler and much better behaved than an Arcanum.

We will first give the definition of a Sanctum and then give the postulates it is built on.

## Definition

A **Sanctum** is a Class equipped with only multiplication which can be partitioned into two disjoint subclasses which form an Outer Sanctum and an Inner Sanctum such that the Sanctum as a whole obeys the basic assumption, inner assumptions, outer assumptions, and the synthetic assumption.

An **Outer Sanctum** is a Class equipped with multiplication which only contains exofinite numbers such that the basic assumption and outer assumptions hold.

An **Inner Sanctum** is a Class equipped with multiplication which only contains subfinite numbes such that the basic assumption and inner assumptions hold.

## The Basic Assumption

**Basic Assumption.** Multiplication is a binary function that is closed.

## Advanced Assumptions

### Outer Assumptions

**Advanced Assumption I.** There exists a number, The Honorary Zeroid, also called Zero, and denoted 0.

**Advanced Assumption II.** There exist numbers, called exofinite numbers, such that, for all exofinite numbers, an exofinite number multiplied by the honorary zeroid results in the honorary zeroid.

**Advanced Assumption III.** The Honorary Zeroid is an exofinite number.

**Advanced Assumption IV.** The product of two exofinite numbers is exofinite.

### Inner Assumptions

**Advanced Assumptions V.** There exists subfinite numbers which are not part of the exofinite numbers

**Advanced Assumption VI.** The product of two subfinite numbers is subfinite.

**Advanced Assumption VII.** Subfinite Numbers associate and commute amongst themselves.

### The Synthetic Assumption

**Advanced Assumption VIII.** The product of an exofinite number and a subfinite number is subfinite.

## Extensions

A Sanctum may be extended by defining division and exponentiation and providing basic order of operations. A Sanctum extended in this way is a **Sanctified Arithmetic**. If an order is imposed on it's elements it's an **Ordered Sanctum**, and if it's ordered and extended then it's an **Orderly Sanctified Arithmetic**.

## Instructive Examples

The original elementary Zeroid Theory forms a Sanctum. Let the Outer Sanctum consist of the real numbers, and let zero be the Honorary Zeroid. Multiplication of real numbers is unique, closed, associative and commutative, thus satisfying the basic assumptions. We also have an Honorary Zeroid, and the set of real numbers multiplied by it always result in zero, thus they can all be classified as exofinite numbers. 0\*0=0 which means 0 is an exofinite number as well. Lastly the product of two real numbers is another real number, thus the product of any two exofinite numbers is exofinite. We take the real numbers, along with multiplication and 0 as the honorary Zeroid to form our Outer Sanctum.

Next we form an Inner Sanctum that consists of all \_a where a is an ordinal number greater than 0. These form our subfinite number class. The product of any two underscores is another underscore under the following rule: \_a \* \_b = \_max(a,b), where max(a,b) returns the larger of the two ordinals. Thus the product of any two subfinite numbers is subfinite.

Now to form our sanctum we only need to ensure that the Synthetic Assumption holds when we combine exofinite and subfinite numbers. For this we use the simple rule, that if x is an exofinite number, and we multiply it by any zeroid, \_a, then we have x \* \_a = \_a. It should be noted that this applies even when we perform 0 \* \_a because 0 is an exofinite number as well.

This forms the basic theory of Zeroids. We may call this *The* *Sanctuary of Simplex Zeroids over the reals*. Note that we may replace the outer sanctum with any other set which has a universal annihilator. So for example we can replace the reals with the complex numbers, or quaternions, or octonions, or sedenions, etc. Each of these has a zero element that acts as a universal annihilator. Note that in the case of reals, complex numbers, quaternions, and octonions, we have the additional nice property that the product of any two non-zero elements is non-zero. This is not true for the sedenions, however this does not violate any of the Sanctuary Assumptions since even in this case, if 0 is the honorary zeroid, every sedenion multiplied by the honorary zeroid is 0, and since 0 is an exofinite number it doesn't matter if two non-zero sedenions multiply to zero.

It should be noted that this is only one possible theory of subfinite numbers that can be created within the Sanctuary framework. Others are easily possible. For example we can create an Inner Sanctum whose basis elements are \_,\_2,\_3,... for all ordinals, ordered in the usual way, however when we multiply them we don't get the smaller of the two basis elements but a new element which is always smaller than both of them. So for example \_\*\_ = \_^2 where \_^2 is smaller than \_. This works as long as they are all subfinite numbers so that the products of any two subfinite numbers is always subfinite. We may sort them so that we get zeroid products of the form \_^x1\*(\_2)^x2\*(\_3)^x3, and they are sorted so that the one with the highest zeroid is always smallest. If they have the same highest zeroid then we check for highest exponents. If these are the same we check the next smallest zeroid and power and so on. We can still insist that the product of an exofinite number and a subfinite number equals the subfinite number. This leads to a much richer structure.

There are even richer structures possible. For example we may include the macro-zeroids among the Outer Sanctum. That is, it's not necessary for there to only be one annihilator, as long as we don't insist that annihilator is absolute. For example we can have the following additional element added to the set of real numbers with the following properties:

x \* \_(-1) = \_(-1) : x != 0

\_(-1) \* \_(-1) = \_(-1)

\_(-1) \* 0 = 0

Since multiplying it by 0 results in zero its an exofinite number, and since its not equal to zero it is larger than 0. Because of this, no matter what we chose as our Inner Sanctum, multiplying \_(-1) by a subfinite number will result in a subfinite number.

We can even add an entire class of macro-zeroids with \_(-a) for any ordinal a>0.

In this way the Sanctum simply provides an open framework to explore different possible Zeroid Theories. Sanctuaries are very safe and well behaved. The main drawback of sanctuaries is that they only involve multiplication. In order to incorporate addition we need an Arcanum.