By Mao L.

This monograph is inspired with surveying arithmetic and physics through CC conjecture, i.e., a mathematical technological know-how will be reconstructed from or made by way of combinatorialization. subject matters coated during this publication contain basic of mathematical combinatorics, differential Smarandache n-manifolds, combinatorial or differentiable manifolds and submanifolds, Lie multi-groups, combinatorial significant fiber bundles, gravitational box, quantum fields with their combinatorial generalization, additionally with discussions on basic questions in epistemology. All of those are worthy for researchers in combinatorics, topology, differential geometry, gravitational or quantum fields.

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Its truth or false can be only decided by logic inference, independent on one knowing it or not. A norm inference is called implication. , if p then q, is a proposition that is false when p is true but q false and true otherwise. There are three propositions related with p → q, namely, q → p, ¬q → ¬p and ¬p → ¬q, called the converse, contrapositive and inverse of p → q. Two propositions are called equivalent if they have the same truth value. It can be shown immediately that an implication and its contrapositive are equivalent.

1 Let (A ; ◦) be an associative system. Then for a1 , a2 , · · · , an ∈ A , the product a1 ◦a2 ◦· · ·◦an is uniquely determined and independent on the calculating order. Proof The proof is by induction. 2 Algebraic Systems (· · · ((a1 ◦ a2 ) ◦ a3 ) ◦ · · ·) ◦ an . If n = 3, the claim is true by deﬁnition. Assume the claim is true for any integers n ≤ k. We consider the case of n = k + 1. , ◦ = 1 . 2 Apply the inductive assumption, we can assume that = (· · · ((a1 ◦ a2 ) ◦ a3 ) ◦ · · ·) ◦ al 1 and = (· · · ((al+1 ◦ al+2 ) ◦ al+3 ) ◦ · · ·) ◦ ak+1 .

We know results for associative and commutative systems following. 1 Let (A ; ◦) be an associative system. Then for a1 , a2 , · · · , an ∈ A , the product a1 ◦a2 ◦· · ·◦an is uniquely determined and independent on the calculating order. Proof The proof is by induction. 2 Algebraic Systems (· · · ((a1 ◦ a2 ) ◦ a3 ) ◦ · · ·) ◦ an . If n = 3, the claim is true by deﬁnition. Assume the claim is true for any integers n ≤ k. We consider the case of n = k + 1. , ◦ = 1 . 2 Apply the inductive assumption, we can assume that = (· · · ((a1 ◦ a2 ) ◦ a3 ) ◦ · · ·) ◦ al 1 and = (· · · ((al+1 ◦ al+2 ) ◦ al+3 ) ◦ · · ·) ◦ ak+1 .