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

Extensive categories, commutative semirings and Galois theory

We describe the Galois theory of commutative semirings as a Boolean Galois theory in the sense of Carboni and Janelidze. Such a Galois structure then naturally suggests an extension to commutative semirings of the classical theory of quadratic equations over commutative rings. We show, however, that our proposed generalization is impossible for connected commutative semirings which are not rings, leading to the conclusion that for the theory of quadratic equations, “minus is needed”. Finally, by considering semirings B which have no non-trivial additive inverses and no non-trivial zero divisors, we present an example of a normal extension of commutative semirings which has an underlying B-semimodule structure isomorphic to B×B

Tensor categorical foundations of algebraic geometry

Tannaka duality and its extensions by Lurie, Sch\"appi et al. reveal that many schemes as well as algebraic stacks may be identified with their tensor categories of quasi-coherent sheaves. In this thesis we study constructions of cocomplete tensor categories (resp. cocontinuous tensor functors) which usually correspond to constructions of schemes (resp. their morphisms) in the case of quasi-coherent sheaves. This means to globalize the usual local-global algebraic geometry. For this we first have to develop basic commutative algebra in an arbitrary cocomplete tensor category. We then discuss tensor categorical globalizations of affine morphisms, projective morphisms, immersions, classical projective embeddings (Segre, Pl\"ucker, Veronese), blow-ups, fiber products, classifying stacks and finally tangent bundles. It turns out that the universal properties of several moduli spaces or stacks translate to the corresponding tensor categories.Comment: PhD thesis; 247 page

Normal Forms for Elements of the ∗{}^*-Continuous Kleene Algebras K⊗RC2′K\mathop{\otimes_{\cal R}} C_2'

The tensor product $K \mathop{\otimes_{\cal R}} C_2'$ of the ${}^*$-continuous Kleene algebra $K$ with the polycyclic ${}^*$-continuous Kleene algebra $C_2'$ over two bracket pairs contains a copy of the fixed-point closure of $K$: the centralizer of $C_2'$ in $K \mathop{\otimes_{\cal R}} C_2'$. We prove a representation of elements of $K\mathop{\otimes_{\cal R}} C_2'$ by automata \`a la Kleene and refine it by normal form theorems that restrict the occurrences of brackets on paths through the automata. This is a foundation for a calculus of context-free expressions. We also show that $C_2'$ validates a relativized form of the ``completeness property'' that distinguishes the bra-ket ${}^*$-continuous Kleene algebra $C_2$ from the polycyclic one.Comment: 32 pages, 4 figure

Monoid Properties as Invariants of Toposes of Monoid Actions

We systematically investigate, for a monoid $M$, how topos-theoretic properties of $\mathbf{PSh}(M)$, including the properties of being atomic, strongly compact, local, totally connected or cohesive, correspond to semigroup-theoretic properties of $M$.Comment: 41 page