By Jorge Almeida
Encouraged via purposes in theoretical laptop technology, the idea of finite semigroups has emerged lately as an self sustaining quarter of arithmetic. It fruitfully combines tools, rules and buildings from algebra, combinatorics, good judgment and topology. merely, the speculation goals at a class of finite semigroups in sure sessions referred to as "pseudovarieties". The classifying features have either structural and syntactical features, the final connection among them being a part of common algebra. along with delivering a foundational research of the idea within the atmosphere method of finite semigroups. This contains learning (relatively) loose and profinite unfastened semigroups and their displays. The recommendations used are illustrated in a scientific examine of varied operators on pseudovarieties of semigroups.
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Example text
We suppose of course that π (b)ϕ = ξψ (b)ϕ if b is in BF and ϕ is in V . Let A be an isomorphism of V with V such that Aπ(g) = π (g)A for all g . Let L be the linear functional L(ϕ) = Aϕ(1) on V . Then a 0 0 1 L π ϕ = Aϕ(a) so that A is determined by L. If we could prove the existence of a scalar λ such that L(ϕ) = λϕ(1) it would follow that Aϕ(a) = λϕ(a) for all a such that Aϕ = λϕ. This equality of course implies the theorem. Observe that L π 1 x 0 1 ϕ =π 1 x 0 1 Aϕ(1) = ψ(x)L(ϕ). 1) Thus we need the following lemma.
It follows from the previous lemma that a 0 0 1 WΦ is zero for almost all a if fΦ is 0. Since WΦ a 0 0 1 is a locally constant function on F × it must vanish everywhere. We have incidentally proved the following lemma. 3 Suppose |µ1 ( )µ−1 2 ( )| = | | with s > −1 and W belongs to W (µ1 , µ2 ; ψ). If W a 0 0 1 =0 for all a in F × then W is 0. 3 Let µ1 and µ2 be two quasi-characters of F × . −1 (i) If neither µ1 µ−1 2 nor µ1 µ2 is αF the representations ρ(µ1 , µ2 ) and ρ(µ2 , µ2 ) are equivalent and irreducible.
Because of (i) we need only show that every linear transformation of X which commutes with all the operators Cn (ν) is a scalar. Suppose T is such an operator. If ϕ belongs to V let Tϕ be the function from F × to X defined by T ϕ(a) = T ϕ(a) . Observe that T ϕ is still in V . This is clear if ϕ belongs to V0 and if ϕ = π(w)ϕ0 we see on examining the Mellin transforms of both sides that T ϕ = π(w)T ϕ0 . Since V = V0 + π(w)V0 the observation follows. T therefore defines a linear transformation of V which clearly commutes with the action of any g in PF .