Rule

Description

Framework: Type 1 Uninomial Unbounded Conducement Sequences

Remark 5.1: ST: Sub-Term; SST: Sub-sub-term; SSST: Sub-sub-sub-term

5.1

Each sequence at a hierarchy is unbounded and the identity of its terms is identified by the ordinal of the terms (e.g. 3^rd).

5.2

Generation G = 1 (d = 0): each term is composed of one term in its internal structure

5.3

Generation G = 2 (d = 1): each term has as many ST as its ordinal number, e.g. Term 3 has 3 sub-terms.

5.4

Generation G = 3 (d = 2): each ST has as many SST as its ordinal number.

5.5

Generation G = 4 (d = 3): each SST has as many SSST as its ordinal.

5.6

Rules (5.1)-(5.6) are recursive with the effect of nesting within each hierarchy the structures of the numbers at its lower hierarchy―the regeneration product rule.

Generalisation: Types 2, 3 and 4 Unbounded Conducemental Sequences

5.7

Type 1 uninomial sequences all have the kernel of: $\langle 1\rangle$ but the kernel size of Types 2, 3 and 4 parent sequences have the kernel: $\langle \begin{array}{ccc}{T}_1& T_2& \begin{array}{ccc}{T}_3& \cdots & T_{\omega }\end{array}\end{array}\rangle$ , with their sizes, $\omega$ , may be less than, equal to and greater than the degree of parent sequences.

5.8

The first term of the parent sequence will replace $\langle 1\rangle$ with $T_1$ , the second term will replace that with: $T_1+T_2$ ; the third term with: $T_1+T_2+T_3$ , $\cdots$ and the ${\omega }^{th}$ term with: $T_1+T_2+T_3+\cdots +T_{\omega }$ . Thereafter, the kernel will be replicated as a whole, i.e. for ${\left(\omega +1\right)}^{th}$ term, there is one additional term of $T_1+T_2+T_3+\cdots +T_{\omega }$ and for ${\left(\omega +2\right)}^{th}$ term, there are two additional terms of $T_1+T_2+T_3+\cdots +T_{\omega }$ and so on.

Remark 5.2: Attention is drawn to the deep level of replication in this rule table.