By Nico F. Benschop

ISBN-10: 1402098294

ISBN-13: 9781402098291

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

teachers and scholars at technical universities, in electric engineering, laptop technological know-how and utilized arithmetic departments, attracted to new advancements of modeling and designing electronic networks (DN: country machines, sequential and combinational good judgment) regularly, as a mixed math/engineering self-discipline. As historical past 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 ideas and their engineering interpretation are brought in a pragmatic type with examples. the inducement in essence is: the significance of the unifying associative algebra of functionality composition (viz. semigoup concept) for the sensible characterisation of the 3 major capabilities in pcs, particularly sequential common sense (state-machines), mathematics and combinational (Boolean) logic.

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**Example text**

With unit code-vectors: a = [1, 0, 0], b = [0, 1, 0], a 3 ≡ b2 ≡ c2 ≡ e = [0, 0, 0], d = [0, 0, 1], and a 2 ≡ a −1 = [2, 0, 0], bd ≡ db = [0, 1, 1]. Component-wise addition produces direct product G = C3 × C2 × C2 of order 12 as reference, and a i bj ck has code [i, j, k] mod (3, 2, 2). Mapping γ : C3 → aut(C2 × C2 ) models the non-commutative behavior of AG4 . In a sequential composition xy this coupling x1 to [x2 , y2 ] is active if x1 ≡ / 0 mod 3, and disabled if x1 ≡ 0 mod 3. The four codes [x2 , x3 ] of C2 × C2 are [1, 0] [0, 1] [1, 1] [0, 0] or in compact notation: 10, 01, 11, 00, with component-wise addition mod 2.

Proof A semigroup S can act as state set Q = S for its representation by a state machine if the square |S| × |S| composition table of S has distinct columns, so for each pair x ≡ / y ∈ S: qx = qy ∈ S for some q ∈ S. If equal columns do occur, as for instance in a left-copy semigroup: ab ≡ a for all a, b ∈ S then only one extra state q0 (as left identity) suffices to obtain distinct columns, while preserving semigroup structure, by defining q0 x ≡ x for all x ∈ S, with Q = {S, q0 } hence |Q| = |S| + 1.

The product ef ≡ g of two idempotents need not be idempotent, but g ∈ Ge Gf . So equivalent maximal subgroups yield under set composition a direct product L × R image of S. (c) =⇒ (d): Consider idempotent a and subset aSa = {axa, (x ∈ S)}. Let the idempotent generated by axa be c ≡ (axa)p with period p. Since c begins and ends with idempotent a, we have ac ≡ ca ≡ c, meaning a ≥ c, and in fact a ≡ c, since no strict ordering occurs in a constant rank semigroup. Hence (axa)p ≡ a : axa generates idempotent a for each x, and is thus in Ga .

### Associative Digital Network Theory An Associative Algebra Approach to Logic, Arithmetic and State Machines by Nico F. Benschop

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