1,258 research outputs found
Ideal Independence, Free Sequences, and the Ultrafilter Number
We make use of a forcing technique for extending Boolean algebras. The same
type of forcing was employed in [BK81], [Kos99], and elsewhere. Using and
modifying a lemma of Koszmider, and using CH, we obtain an atomless BA, A such
that f(A) = smm(A) < u(A), answering questions raised by [Mon08] and [Mon11]
Syntactic Monoids in a Category
The syntactic monoid of a language is generalized to the level of a symmetric
monoidal closed category D. This allows for a uniform treatment of several
notions of syntactic algebras known in the literature, including the syntactic
monoids of Rabin and Scott (D = sets), the syntactic semirings of Polak (D =
semilattices), and the syntactic associative algebras of Reutenauer (D = vector
spaces). Assuming that D is an entropic variety of algebras, we prove that the
syntactic D-monoid of a language L can be constructed as a quotient of a free
D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the
transition D-monoid of the minimal automaton for L in D. Furthermore, in case
the variety D is locally finite, we characterize the regular languages as
precisely the languages with finite syntactic D-monoids
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