By Nico F. Benschop

*Associative electronic community Theory* is meant for researchers at commercial laboratories,

teachers and scholars at technical universities, in electric engineering, computing device technological know-how and utilized arithmetic departments, drawn to new advancements of modeling and designing electronic networks (DN: kingdom machines, sequential and combinational common sense) in most cases, as a mixed math/engineering self-discipline. As history an undergraduate point of contemporary utilized algebra (Birkhoff-Bartee: smooth utilized Algebra - 1970, and Hartmanis-Stearns: Algebraic constitution of Sequential Machines - 1970) will suffice.

Essential thoughts and their engineering interpretation are brought in a pragmatic model with examples. the inducement in essence is: the significance of the unifying associative algebra of functionality composition (viz. semigoup conception) for the sensible characterisation of the 3 major features in pcs, specifically sequential good judgment (state-machines), mathematics and combinational (Boolean) logic.

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**Example text**

Denote the direct product Cq × Cq × · · · × Cq (k times) briefly as [Cq ]k , and let q be prime. Using a k-component code, the k unit vectors form a necessary and sufficient set of generators, so dim([Cq ]k ) = k while deg([Cq ]k ) = kq. Recall [0, 1] and [1, 0] generate [C2 ]2 , or alternatively [0, 1] and [1, 1], or [1, 0] and [1, 1]. Generalizing this, any subset of k non-zero codes that covers each unit vector or some non-zero multiple of it (any one of the q − 1 non-zero elements of Cq generates Cq ) also generates [Cq ]k .

3 Recursion (RFG) generates all full groups for n > 2. The structure of FGn derives from that of FGn−1 , because defining r-congruence δ, specified by any stabilizer FGn−1 of FGn , is extended by subgroup Cn as rightcomposing elements, functioning as last (rightmost) n-counter code-component, Cn ∼ = Z(+) mod n. Recall an even [odd] permutation is obtained from e by swapping an even [odd] number of state pairs. Their composition is like addition mod 2, thus Z2 (+) with odd = 1, even = 0 which is isomorphic to a 2-cycle C2 .

Hence its automorphism group is FG3 with C3 as subgroup representing a 3-cycle rotation α of the three non-zero (b, d, bd) with α 3 ≡ α 0 ≡ the identity transformation, for instance in compact notation: α : 01 → 10 → 11 → 01 and α(00) = 00 (fixing code 00), with: coupling map γ : x1 = {1, 2, 0} → {α, α , } = C3 ⊂ aut(C2 × C2 ). 2 44 3 Coupling State Machines The 3-component code for group AG4 , over 3 + 2 + 2 = 7 states, yields the next arithmetic model of its cascade decomposition AG4 = C3 |> (C2 × C2 ): [x1 , x2 , x3 ][y1 , y2 , y3 ] → [z1 , z2 , z3 ], where: z1 ≡ x1 + y1 mod 3, coupled by x1 (mod 3) to: z2 ≡ α (x2 + y2 ) mod 2, x1 z3 ≡ α x1 (x3 + y3 ) mod 2.