Let A be the affine plane over a field K of characteristic 0. Birational morphisms of A are mappings A → A given by polynomial mappings φ of the polynomial algebra K[x, y] such that for the quotient fields, one has K(φ(x), φ(y)) = K(x, y). Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping τx given by x → x, y → xy. For a while, it was an open...

A skew morphism of a finite group B is permutation φ that preserves the identity element and has property for every ∈ there exists positive integer ia such φ(ab) = φ(a)φia(b) all b B. The problem classifying morphisms cyclic groups notoriously hard, with no classification available up to date. Each Zn closely related specific Z|⟨φ⟩|, called quotient φ. In this paper, we use relationship other o...

1 is unique (1 = 11 = 1, so any left identity equals any right identity). A morphism takes 1 to the identity of φX (since φ(x) = φ(1x) = φ(1)φ(x)), but φ(1) = 1 is an extra property that ought to be satisfied by morphisms; its kernel is the sub-algebra φ(1); zero object is { 1 }; not Cartesian-closed (the terminal and initial objects are the same, yet not all groups are isomorphic). A quasi-gro...

Let E = {E1, . . . , Ek} be a set of regular languages over a finite alphabet Σ. Consider morphism φ : ∆ → (S, ·) where ∆ is the semigroup over a finite set ∆ and (S, ·) = 〈E〉 is the finitely generated semigroup with E as the set of generators and language concatenation as a product. We prove that the membership problem of the semigroup S, the set [u] = {v ∈ ∆ | φ(v) = φ(u)}, is a regular langu...

Let ~ C be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category ~ C is called ~ C-closed if for each morphism Φ ⊂ X×Y in the category ~ C the image Φ(X) = {y ∈ Y : ∃x ∈ X (x, y) ∈ Φ} is closed in Y. In the paper we survey existing and new results on topological groups, which are ~...

Using the closed point sieve, we extend to finite fields the following theorem proved by A. Bhatnagar and L. Szpiro over infinite fields: if X is a closed subscheme of P over a field, and φ : X → X satisfies φOX(1) ' OX(d) for some d ≥ 2, then there exists r ≥ 1 such that φ extends to a morphism P → P.

For all integers r ≥ 2 and any smooth and connected projective curve X, let ρX(r) denote the minimal integer d such that there is a morphism φ : X → P birational onto its image and such that deg(φ(X)) = d and φ(X) spans Pr. Fix integers d, g such that d ≥ 8 and d2/6 < g ≤ d2/4−d. Here we prove the existence of a smooth genus g curveX such that ρX(3) = d. Mathematics Subject Classification: 14H5...

Using the closed point sieve, we extend to finite fields the following theorem proved by A. Bhatnagar and L. Szpiro over infinite fields: if X is a closed subscheme of P over a field, and φ : X → X satisfies φOX(1) ' OX(d) for some d ≥ 2, then there exists r ≥ 1 such that φ extends to a morphism P → P.

This short, expository note proves existence of the maximal quotient of a variety by free rational curves. 1. Definition of a maximal free rational quotient Definition 1.1. Let V be a Deligne-Mumford stack over a field k, and denote the smooth locus by V sm ⊂ V . A 1-morphism f : Pk → V sm is a free rational curve to V if fTV is generated by global sections and has positive degree. Let S be an ...

We consider stacks of filtered φ-modules over rigid analytic spaces and adic spaces. We show that these modules parametrize p-adic Galois representations of the absolute Galois group of a p-adic field with varying coefficients over an open substack containing all classical points. Further we study a period morphism (defined by Pappas and Rapoport) from a stack parametrising integral data and de...