Download e-book for kindle: Structural theory of automata, semigroups, and universal by M. Goldstein, Valery B. Kudryavtsev, Ivo G. Rosenberg
By M. Goldstein, Valery B. Kudryavtsev, Ivo G. Rosenberg
Numerous of the contributions to this quantity deliver ahead many collectively invaluable interactions and connections among the 3 domain names of the name. constructing them used to be the most goal of the NATO ASI summerschool held in Montreal in 2003. even though a few connections, for instance among semigroups and automata, have been recognized for a very long time, constructing them and surveying them in a single quantity is novel and with a bit of luck stimulating for the long run. one other element is the emphasis at the structural idea of automata that stories how you can build immense automata from small ones. the quantity additionally has contributions on most sensible present study or surveys within the 3 domain names. One contribution even hyperlinks clones of common algebra with the computational complexity of machine technology. 3 contributions introduce the reader to investigate within the former East block.
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Extra info for Structural theory of automata, semigroups, and universal algebra
5 Proposition Let S be a pro-V semigroup and let K ⊆ S. Then the following conditions are equivalent: (1) there exists a continuous homomorphism ϕ : S → F such that F ∈ V and K = ϕ−1 ϕK; (2) K is clopen; (3) the syntactic congruence ρK is clopen. In particular, all these conditions imply that the syntactic semigroup Synt K belongs to V. Proof Assuming the existence of a function ϕ satisfying (1), we deduce that K is clopen since it is the inverse image under a continuous function of a clopen set.
Then it is routine to check that ϕ is a continuous homomorphism satisfying the required conditions. This establishes the equivalence (1) ⇔ (2). If K is clopen then ρK is clopen by Hunter’s Lemma. This proves (2) ⇒ (3) and for the converse it suﬃces to recall that K is saturated by ρK . Finally, assuming (1), K is recognized by a semigroup from V. 4, Synt K belongs to V since V is closed under taking divisors. ✷ We note that the assumption that Synt K belongs to V for a subset K of a pro-V semigroup S does not suﬃce to deduce that K is clopen, as the following example shows.
Proﬁnite semigroups and applications 25 In the case n = 1, take ϕ ∈ End Ω1 S determined by the 1-tuple (xm ). Then ϕ has an ω ω-power in End Ω1 S and we may consider the operation ϕω (x) which we denote xm since ω k! xm = ϕω (x) = lim ϕk! (x) = lim xm . 15 Examples (1) It is now an easy exercise, which we leave to the reader, to show that, for a prime p, ω Gp = [[xp = 1]]. (2) Let Gnil denote the pseudovariety of all ﬁnite nilpotent groups. Consider the endomorphism ϕ ∈ End Ω2 S deﬁned by the pair ([x, y], y) where [x, y] denotes the commutator deﬁned earlier.
Structural theory of automata, semigroups, and universal algebra by M. Goldstein, Valery B. Kudryavtsev, Ivo G. Rosenberg