Schützenberger Products in a Category

Abstract

The Schützenberger product of monoids is a key tool for the algebraic treatment of language concatenation. In this paper we generalize the Schützenberger product to the level of monoids in an algebraic category D, leading to a uniform view of the corresponding constructions for monoids (Schützenberger), ordered monoids (Pin), idempotent semirings (Klíma and Polák) and algebras over a field (Reutenauer). In addition, assuming that D is part of a Stone-type duality, we derive a characterization of the languages recognized by Schützenberger products.Liang-Ting Chen acknowledges support from AFOSR. Henning Urbat acknowledges support from DFG under project AD / -. arXiv:1605.01810v1 [cs.FL] 6 May 2016 Schützenberger Products in a Category theorems for regular languages. See also [ , ] for related duality-based work. Recently, Bojańczyk [ ] proposed to use monads instead of monoids to get a categorical grasp on languages beyond finite words. By combining this idea with our duality framework, we established in [ , ] a variety theorem that covers most Eilenberg-type correspondences known in the literature, e.g. for languages of finite words, infinite words, words on linear orderings, trees, and cost functions. PreliminariesIn this paper we study monoids and language recognition in algebraic categories. The reader is assumed to be familiar with basic universal algebra and category theory; see the Appendix for a toolkit. We call a variety D of algebras or ordered algebras commutative if, for any two algebras A, B ∈ D, the set [A, B] of morphisms from A to B forms an algebra of D with operations taken pointwise in B. Our applications involve the commutative varieties Set (sets), Pos (posets, as ordered algebras without any operation), JSL (join-semilattices with 0), K-Vec (vector spaces over a field K) and S-Mod (modules over a commutative semiring S with 0, 1). Note that JSL and K-Vec are special cases of S-Mod for S = {0, 1}, the two-element semiring with 1 + 1 = 1, and S = K, respectively.Notation . . Let A , B, C , D always denote commutative varieties of algebras or ordered algebras. We write Ψ = Ψ D : Set → D for the left adjoint to the forgetful functor |−| : D → Set; thus Ψ X is the free algebra of D over X. For simplicity, we assume that X is a subset of |Ψ X| and the universal map X |Ψ X| is the inclusion. Denote by 1 D = Ψ 1 the free one-generated algebra. Example . . ( ) For D = Set or Pos we have Ψ X = X (discretely ordered). ( ) For D = JSL we get Ψ X = (P f X, ∪), the semilattice of finite subsets of X. ( ) For D = S-Mod we have Ψ X = S (X) , the S-module of all finite-support functions X → S with sum and scalar product defined pointwise. Definition . . Let A, B, C ∈ D. By a bimorphism from A, B to C is meant a function f : |A| × |B| → |C| such that the maps f (a, −) : |B| → |C| and f (−, b) : |A| → |C| carry morphisms of D for every a ∈ |A| and b ∈ |B|. A tensor product of A and B is a universal bimorphism t A,B : |A| × |B| → |A ⊗ B|, in the sense that for any bimorphism f : |A|×|B| → |C| there is a unique f...

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“…Also, the duality-theoretic account in Section 6 leads to a Reutenauer-type characterisation theorem, akin to the one in Gehrke et al (2016). It would be interesting to identify a common framework for our contributions and the recent work (Chen and Urbat, 2016).…”

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Quantifiers on languages and codensity monads

“…The triangle on the left commutes by the leftmost diagram in (7). To show that the other triangle commutes,…”