By Garrett P.

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Finite Semigroups and Universal Algebra (Series in Algebra, Vol 3)

Encouraged by way of functions in theoretical desktop technological know-how, the speculation of finite semigroups has emerged in recent times as an self sustaining zone of arithmetic. It fruitfully combines equipment, principles and structures from algebra, combinatorics, common sense and topology. only, the speculation goals at a type of finite semigroups in definite periods referred to as "pseudovarieties".

Additional info for Algebras and Involutions(en)(40s)

Example text

D−1 } is a K-basis for D, and since π i ∈ k if and only if i = 0 mod d, for 1 ≤ i < d we have (full trace of left multiplication by απ i on D = 0 Thus, the reduced trace is also 0 on such elements. On the other hand, the restriction of the reduced trace on D to K is equal to the Galois trace from K to k. For x ∈ D, we may write αi π −i x= 0≤i

If D = k we are done. This leaves the unique quaternion division algebra D to be considered. The case that the involution θ on the quaternion division algebra D is of first kind is easy, since we already know that D has a main involution, so by Skolem-Noether any other involution of first kind differs by a conjugation. Now suppose that θ is of second kind. Let \alf → α be the main involution. Then α → (αθ ) is an automorphism of order 2 of D and gives a non-trivial automorphism τ on k. The set Do = {x ∈ D : xθ } is a subring of D containing the subfield ko of k fixed by τ .

Since the field P/P is finite, there is an integer m such that for λ( ) x = Then define ordα α−1 = m = xp mod P mod P2 αα−1 = (q − 1) =− mod P2 µ∈M So π is also a local parameter, λ(π) = λ( ), and π normalizes M . /// Corollary: With M and π as in the theorem, every x ∈ D× has a unique expression of the form αi π i x= i≥m with αi ∈ M , and where m is the uniquely determined integer so that x ∈ πm · O× Proof: If x ∈ O× , then αo = ω(x) ∈ M satisfies α = x mod P, and by the theorem is uniquely determined in M by this property.