Effective dimension of finite semigroups

In this paper we discuss various aspects of the problem of determining the minimal dimension of an injective linear representation of a finite semigroup over a field. We outline some general techniques and results, and apply them to numerous examples.Comment: To appear in J. Pure Appl. Al

Equivariant Cohomology of Rationally Smooth Group Embeddings

We describe the equivariant cohomology ring of rationally smooth projective embeddings of reductive groups. These embeddings are the projectivizations of reductive monoids. Our main result describes their equivariant cohomology in terms of roots, idempotents, and underlying monoid data. Also, we characterize those embeddings whose equivariant cohomology ring is obtained via restriction to the associated toric variety. Such characterization is given in terms of the closed orbits.Comment: 25 pages. Final version. To appear in Transformation Group

Some remarks on blueprints and F1{\mathbb F}_1-schemes

Over the past two decades several different approaches to defining a geometry over ${\mathbb F}_1$ have been proposed. In this paper, relying on To\"en and Vaqui\'e's formalism, we investigate a new category ${\mathsf{Sch}}_{\widetilde{\mathsf B}}$ of schemes admitting a Zariski cover by affine schemes relative to the category of blueprints introduced by Lorscheid. A blueprint, that may be thought of as a pair consisting of a monoid $M$ and a relation on the semiring $M \otimes_{{\mathbb F}_1} \mathbb N$, is a monoid object in a certain symmetric monoidal category $\mathsf B$, which is shown to be complete, cocomplete, and closed. We prove that every $\widetilde{\mathsf B}$-scheme $\Sigma$ can be associated, through adjunctions, with both a classical scheme $\Sigma_{\mathbb Z}$ and a scheme $\underline{\Sigma}$ over ${\mathbb F}_1$ in the sense of Deitmar, together with a natural transformation $\Lambda\colon \Sigma_{\mathbb Z}\to \underline{\Sigma}\otimes_{{\mathbb F}_1} {\mathbb Z}$. Furthermore, as an application, we show that the category of "${\mathbb F}_1$-schemes" defined by A. Connes and C. Consani can be naturally merged with that of $\widetilde{\mathsf B}$-schemes to obtain a larger category, whose objects we call "${\mathbb F}_1$-schemes with relations".Comment: 39 pages. Revised final version to appear in S\~ao Paulo Journal of Mathematical Sciences, with the addition of an Appendix on "Fibered categories and stacks

Monoid varieties with extreme properties

Finite monoids that generate monoid varieties with uncountably many subvarieties seem rare, and surprisingly, no finite monoid is known to generate a monoid variety with countably infinitely many subvarieties. In the present article, it is shown that there are, nevertheless, many finite monoids that generate monoid varieties with continuum many subvarieties; these include any finite inherently non-finitely based monoid and any monoid for which $xyxy$ is an isoterm. It follows that the join of two Cross monoid varieties can have a continuum cardinality subvariety lattice that violates the ascending chain condition. Regarding monoid varieties with countably infinitely many subvarieties, the first example of a finite monoid that generates such a variety is exhibited. A complete description of the subvariety lattice of this variety is given. This lattice has width three and contains only finitely based varieties, all except two of which are Cross

Monoid Embeddings of Symmetric Varieties

We determine when an antiinvolution on an adjoint semisimple linear algebraic group extends to an antiinvolution on a $J$-irreducible monoid. Using this information, we study a special class of compactifications of symmetric varieties. Extending the work of Springer on involutions, we describe the parametrizing sets of Borel orbits in these special embeddings

Using Rewriting Systems to Compute Kan Extensions and Induced Actions of Categories

The basic method of rewriting for words in a free monoid given a monoid presentation is extended to rewriting for paths in a free category given a `Kan extension presentation'. This is related to work of Carmody-Walters on the Todd-Coxeter procedure for Kan extensions, but allows for the output data to be infinite, described by a language. The result also allows rewrite methods to be applied in a greater range of situations and examples, in terms of induced actions of monoids, categories, groups or groupoids.Comment: 31 pages, LaTeX2e, (submitted to JSC

Semi-galois Categories I: The Classical Eilenberg Variety Theory

This paper is an extended version of our proceedings paper announced at LICS'16; in order to complement it, this version is written from a different viewpoint including topos-theoretic aspect on our work. Technically, this paper introduces and studies the class of semi-galois categories, which extend galois categories and are dual to profinite monoids in the same way as galois categories are dual to profinite groups; the study on this class of categories is aimed at providing an axiomatic reformulation of Eilenberg's theory of varieties of regular languages--- a branch in formal language theory that has been developed since the mid 1960's and particularly concerns systematic classification of regular languages, finite monoids, and deterministic finite automata. In this paper, detailed proofs of our central results announced at LICS'16 are presented, together with topos-theoretic considerations. The main results include (I) a proof of the duality theorem between profinite monoids and semi-galois categories, extending the duality theorem between profinite groups and galois categories; based on this results on semi-galois categories, we then discuss (II) a reinterpretation of Eilenberg's theory from a viewpoint of duality theorem; in relation with this reinterpretation of the theory, (III) we also give a purely topos-theoretic characterization of classifying topoi BM of profinite monoids M among general coherent topoi, which is a topos-theoretic application of (I). This characterization states that a topos E is equivalent to the classifying topos BM of some profinite monoid M if and only if E is (i) coherent, (ii) noetherian, and (iii) has a surjective coherent point. This topos-theoretic consideration is related to the logical and geometric problems concerning Eilenberg's theory that we addressed at LICS'16, which remain open in this paper.Comment: Updated some part of our proceedings paper published in Proc. of LICS1

Toric varieties: Simple combinatorial and geometrical structure of multipartite quantum systems

We investigate the geometrical structure of multipartite states based on the construction of toric varieties. We show that the toric variety represents the space of general pure states and projective toric variety defines the space of separable set of multi-qubits states. We also discuss in details the construction of single-, two-, three-, and multi- qubits states. This construction gives a very simple and elegant visual representation of the geometrical structure of multipartite quantum systems.Comment: 14 page

Variants of finite full transformation semigroups

The variant of a semigroup S with respect to an element a in S, denoted S^a, is the semigroup with underlying set S and operation * defined by x*y=xay for x,y in S. In this article, we study variants T_X^a of the full transformation semigroup T_X on a finite set X. We explore the structure of T_X^a as well as its subsemigroups Reg(T_X^a) (consisting of all regular elements) and E_X^a (consisting of all products of idempotents), and the ideals of Reg(T_X^a). Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size.Comment: 25 pages, 6 figures, 1 table - v2 includes a couple more references - v3 changes according to referee comments (to appear in IJAC

Betti numbers of toric varieties and eulerian polynomials

It is well-known that the Eulerian polynomials, which count permutations in $S_n$ by their number of descents, give the $h$-polynomial/$h$-vector of the simple polytopes known as permutohedra, the convex hull of the $S_n$-orbit for a generic weight in the weight lattice of $S_n$. Therefore the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra. In this paper we derive recurrences for the $h$-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a non-generic weight, namely a weight fixed by only the simple reflections $J=\{s_{n},s_{n-1},s_{n-2} \cdots,s_{n-k+2},s_{n-k+1}\}$ for some $k$ with respect to the $A_n$ root lattice. Furthermore, they give rise to certain rationally smooth toric varieties $X(J)$ that come naturally from the theory of algebraic monoids. Using effectively the theory of reductive algebraic monoids and the combinatorics of simple polytopes, we obtain a recurrence formula for the Poincar\'e polynomial of $X(J)$ in terms of the Eulerian polynomials