Abstract
In computer science the Myhill–Nerode Theorem states that a set L of words in a finite alphabet is accepted by a finite automaton if and only if the equivalence relation ∼L, defined as x ∼L y if and only if xz ∈ L exactly when yz ∈ L, ∀z, has finite index. The Myhill–Nerode Theorem can be generalized to an algebraic setting giving rise to a collection of bialgebras which we call Myhill–Nerode bialgebras. In this paper we investigate the quasitriangular structure of Myhill–Nerode bialgebras.
Highlights
Let Σ0 be a finite alphabet and let Σ0 denote the set of words formed from the letters in Σ0
The Myhill–Nerode Theorem of computer science states that L is accepted by a finite automaton if and only if ∼L has finite index
In [3, Theorem 5.4] the authors generalize the Myhill–Nerode theorem to an algebraic setting in which a finiteness condition involving the action of a semigroup on a certain function plays the role of the finiteness of the index of ∼L, while a bialgebra plays the role of the finite automaton which accepts the language
Summary
Let Σ0 be a finite alphabet and let Σ0 denote the set of words formed from the letters in Σ0. A K-bialgebra is a K-vector space B together with maps mB , ηB , ∆B , B for which (B, mB , ηB ) is a K-algebra and (B, ∆B , B ) is a K-coalgebra and for which ∆B and B are algebra homomorphisms. A K-linear map φ : B → B 0 is a bialgebra homomorphism if φ is both an algebra and coalgebra homomorphism. A K-Hopf algebra is a bialgebra H together with an additional K-linear map σH : H → H that satisfies mH (IH ⊗ σH )∆H (h) = H (h)1H = mH (σH ⊗ IH )∆H (h). By Proposition 2.1, B ∗ is an algebra with maps mB ∗ = ∆∗B and ηB ∗ = ∗B. Xn } be a finite semigroup with unity element 1KG = x1 , and let KG denote the semigroup bialgebra.
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