MAGMA

The process of assigning various algebraic structures to verses and their contents begins with an appropriate underlying set of objects at the outset. It is to serve as the foundation for subsequent developments. The set of all verses {1:1, ... , 114:6} thus forms such an ideal and fitting starting place. The objective at the most basic level is then to enrich the set with a binary operation, turning it into a magma. The binary operation allows to combine any two elements from the set using the operation, resulting in another element within the set. The binary operation is generally referred to as the product or the multiplication. A magma can be viewed as the most basic algebraic structure on a given set. It requires no additional rules and restrictions.

To define a product on the set of verses it is instructive to review the usual multiplication of familiar numbers. The multiplication of numbers can be thought of as a rule or a function of two variables, M(⋅, ⋅), taking in two numbers and outputting one number, their product:

M(x, y) = xy.

A function of two variables can equivalently be expressed as a sequence of functions of one variable:

M(x, y) = P_x(y),

where

P_x(y) = xy.

Here, P_x(y) is a sequence of functions indexed by x that takes one input y and returns its product with x. Stated differently, the one-variable function P_x(y) can be thought of as a function that is attached to number x. It takes a number y as input and returns the product xy. This method of converting a function with multiple variables into a sequence of functions of a single variable is called currying. It treats the first variable as a parameter to index the sequence of functions taking one fewer variable.

Similarly, let M(U, V) be the multiplication function to be defined on set of verses that takes an arbitrary pair of verses U and V and returns a verse W as their product, UV. Then, via currying it transforms into a sequence of one-variable functions:

W = UV = M(U, V) = P_U(V),

where P_U(⋅) is a function that maps verses to verses that is defined specifically for verse U. It takes one verse (represented by "⋅") as input and returns another verse as output. Since, the cordisans form a sequence of one-variable functions indexed by verses (each verse has its own distinct cordisan), they are the perfect choice. For an arbitrary verse U let its cordisan be U'. Then, P_U(⋅) = U'(⋅). The product of U and V can now be defined as

W = UV = P_U(V) = U'(V)

Switching the order of U and V gives

VU = P_V(U) = V'(U),

where V'(⋅) is the cordisan of V. Since, the cordisans U'(⋅) and V'(⋅) are entirely different functions, U'(⋅) ≠ V'(⋅), it follows that, in general, U'(V) ≠ V'(U). Consequently, the product is noncommutative: UV ≠ VU.

It can, furthermore, be shown that the product is nonassociative as well. Let U, V and W be three verses. Then, (UV)W = (UV)'W, where (UV)' is the cordisan of UV; the verse that is the product of U and V. The product UV itself is U'(V). Therefore, (UV)W = (U'(V))'(W). The other bracketing U(VW) is obtained similarly: U(VW) = U'(VW) = U'(V'(W)). For the triple product UVW to be associative it is required that (U'(V))'(W) = U'(V'(W)). Since, this must be true for all W, it requires (U'(V))' = U'V'. However, this is clearly inconceivable since (U'(V))' is the cordisan anchored to U'(V) while U'V' is the composition of the cordisan U'(⋅) with V'(⋅) which is totally unrelated to the cordisan at U'(V). Hence, (UV)W ≠ U(VW).

A distinctive characteristic of the multiplication so defined on verses is the lack of the multiplicative neutral element (also called the multiplicative identity or the unit) E such that EV = VE = V for all verses V, that behaves similarly to 1 on the set of numbers, 1n = n1 = n, for all numbers n. This can be readily seen as follows. EV = V implies E'(V) = V for all V. This implies E'(⋅) is the trivial identity function Id(X) = X. However, no cordisan is the identity function. Separately, VE = V implies V'(E) = V for all V. This makes E a verse with exceptionally unique property since it requires all cordisans to map E to the verse they are anchored to. No such restriction exists on any cordisan. Accordingly, no (left or right) identity can exist.

Among all verses in the magma, verse one (1:1) is unique as it partially shares a comparable property to zero. Zero is the unique annihilator under multiplication in the standard number systems: 0 x n = n x 0 = 0, for any n. It has the absorbing property when it is multiplied with any number. Let T = 1:1, then for any verse V, VT = V'(T) = T, since 1:1 is the fixed-point of all cordisans. This make 1:1 the right-absorbing element. However, unlike zero, it is not left-absorbing: TV = T'(V) ≠ T. This is because it would require T'(V) = T for all V, forcing the cordisan T'(⋅) to be a constant function that maps all verses to 1:1. This is impossible as all cordisans are bijective maps. Thus, the absorption property of verse one is directional or one-sided. It behaves exactly like zero when multiplied from the right while behaving like normal elements when multiplying from the left.

The magma can further be enhanced into an algebra over the integers ℤ. The operation of addition is taken to be the formal sum. The elements in the algebra are formal linear combinations of the magma elements: V = c₁V₁ + ... + c_nV_n, with c_i ∈ ℤ and verses V_i in the magma. The multiplication in the magma extends linearly to all elements in the algebra. As an example, consider R = c₁V₁ + c₂V₂ and S = c₃V₃ + c₄V₄. Here, R and S are elements in the algebra with c_i ∈ ℤ and V_i verses in the magma. Then RS = (c₁V₁ + c₂V₂) (c₃V₃ + c₄V₄) = c₁c₃ V₁V₃ + c₁c₄ V₁V₄ + c₂c₃ V₂V₃ + c₂c₄ V₂V₄ with c_ic_j ∈ ℤ and the verse products V_iV_j are evaluated in the magma: V_iV_j = V'_iV_j.

With T = 1:1 as before, given an arbitrary verse V in the magma, let S = V − T be an element in its magma algebra. Then, ST = (V − T)T = VT − TT = T − T = 0. The right absorption property of verse one consequently leads to the presence of zero divisors. That is when the product of two non-zero elements is zero: S ≠ 0 ≠ T, while ST = 0. In general, if S = ∑ c_iV_i such that ∑ c_i = 0, it follows that ST = (∑ c_iV_i)T = ∑ c_iV_iT = ∑ c_iT = (∑ c_i)T = 0T = 0. Thus, any linear combination of magma elements whose coefficients sum to zero becomes a zero-divisor when it is paired with T on the right.

The magma algebra has a richer algebraic structure than the underlying magma since it possess extra features that the original magma lacks: linear structure and distributivity: x(y + z) = xy + xz and (x + y)z = xz + yz. While a magma is a set with a single binary operation, its magma algebra combines that operation with a module structure, creating a more refined platform for further analysis. The set of integers ℤ, however, only forms a ring since elements other than ±1 do not have multiplicative inverses (reciprocals). Replacing ℤ with a field turns the module into a vector space that is better behaved and more suitable to work with. Modules in general have more complex behavior and may not always have basis or well-defined dimension, unlike vector spaces.

The set of all verses is thus equipped with the structure of a magma algebra whose binary product is defined purely in terms of the cordisans of verses. All characteristics of the algebra emerge exclusively from the verses with no additional constraints. It is a non-unital, noncommutative and nonassociative algebra. The singular verse one (1:1) is the unique right-absorbing element in the magma that acts as a catalyst that directly leads to construction of the zero-divisors in the algebra.