A Survey of Schreier-Type Extensions of Monoids

Quotients of monoid extensions and their interplay with Baer sums

Cosetal extensions of monoids generalise extensions of groups, special Schreier extensions of monoids and Leech's normal extensions of groups by monoids. They share a number of properties with group extensions, including a notion of Baer sum when the kernel is abelian. However, unlike group extensions (with fixed kernel and cokernel) there may be nontrivial morphisms between them. We explore the structure of the category of cosetal extensions and relate it to an analogue of second cohomology groups. Finally, the order structure and additive structures are combined to give an indexed family of …

Schreier rewriting beyond the classical setting

Using actions of free monoids and free associative algebras, we establish some Schreier-type formulas involving the ranks of actions and the ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish the generalization of the Schreier formula to the case of subgroups of infinite index. We also study and apply large modules over free associative algebras in the spirit of the paper Olshanskii, A. Yu.; Osin, D.V., Large groups and their periodic quotients, Proc. Amer. Math. S…

Semi-biproducts in monoids

It is shown that the category of semi-biproducts in monoids is equivalent to a category of pseudo-actions. A semi-biproduct in monoids is at the same time a generalization of a semi-direct product in groups and a biproduct in commutative monoids. Every Schreier extension of monoids can be seen as an instance of a semi-biproduct; namely a semi-biproduct whose associated pseudo-action has a trivial correction system. A correction system is a new ingredient that must be inserted in order to obtain a pseudo-action out of a pre-action and a factor system. In groups, every correction system is trivi…

Mixed Characteristic Artin Schreier Polynomials

We present here a version of the Artin-Schreier polynomial that works in any characteristic. Let $C_p$ be the cyclic group of prime order $p$. Equivalently, we prove one can lift degree $p$ cyclic $C_p$ extensions over local rings $R,M$ where $R/M$ has characteristic $p$ and $R$ has arbitrary characterstic. Let $ρ\in \Bbb C$ be a primitive $p$ root of one. We first consider the case $R$ has a primitive $p$ root of one, by which we mean that there is a given homomorphism $f: \Bbb Z[ρ] \to R$. In this context we can write a specific polynomial which generalizes the Artin-Schreier polynomial. We …

近环的扩张理论与半直和

An extension theory and semidirect sums for nearrings

本文提出近环的一种扩张理论，其半直和为特例，近环理论中已有不少相关构造实例，丰富了近环的研究方法。

On totally ramified extensions of discrete valued fields

We give a simple characterization of the totally wild ramified valuations in a Galois extension of fields of characteristic p. This criterion involves the valuations of Artin-Schreier cosets of the F_{p^r}^\times-translation of a single element. We apply the criterion to construct some interesting examples.

Realization-obstruction exact sequences for Clifford system extensions

For every action $ϕ\in\text{Hom}(G,\text{Aut}_k(K))$ of a group $G$ on a commutative ring $K$ we introduce two abelian monoids. The monoid $\text{Cliff}_k(ϕ)$ consists of equivalent classes of $G$-graded Clifford system extensions of type $ϕ$ of $K$-central algebras. The monoid $\mathcal{C}_k{(ϕ)}$ consists of equivariant classes of generalized collective characters of type $ϕ$ from $G$ to the Picard groups of $K$-central algebras. Furthermore, for every such $ϕ$ there is an exact sequence of abelian monoids $$0\to H^2(G,K^*_ϕ)\to\text{Cliff}_k(ϕ)\to\mathcal{C}_k{(ϕ)}\to H^3(G,K^*_ϕ).$$ The ri…

On Serre dimension of monoid algebras and Segre extensions

Let $R$ be a commutative noetherian ring of dimension $d$ and $M$ be a commutative$,$ cancellative$,$ torsion-free monoid of rank $r$. Then $S$-$dim(R[M]) \leq max\{1, dim(R[M])-1 \} = max\{1, d+r-1 \}$. Further$,$ we define a class of monoids $\{\mathfrak{M}_n\}_{n \geq 1}$ such that if $M \in \mathfrak{M}_n$ is seminormal$,$ then $S$-$dim(R[M]) \leq dim(R[M]) - n= d+r-n,$ where $1 \leq n \leq r$. As an application, we prove that for the Segre extension $S_{mn}(R)$ over $R,$ $S$-$dim(S_{mn}(R)) \leq dim(S_{mn}(R)) - \Big[\frac{m+n-1}{min\{m,n\}}\Big] = d+m+n-1 - \Big[\frac{m+n-1}{min\{m,n\}}\…

幺半群的拓扑有限性性质。第2部分：特殊幺半群、单关系幺半群、融合自由积与 HNN 扩张

Topological finiteness properties of monoids. Part 2: special monoids, one-relator monoids, amalgamated free products, and HNN extensions

本文应用拓扑方法证明特殊幺半群从其单位群继承有限性性质，并据此证明单关系幺半群的 Lyndon 恒等式定理，同时发展 Bass-Serre 理论研究融合自由积与 HNN 扩张的闭包性质。

Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology

In this monograph, we extend S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore we establish an explicit description of an isomorphism by A. Neeman and V. Retakh, which links $\mathrm{Ext}$-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the associated Quillen groupoid. As a main result, we show that our construction behaves well with respect to structure preserving functors between exact monoidal categories…

扩张范畴中来自中心的括号与乘积

Brackets and products from centres in extension categories

该研究将Schwede关于结合代数双模上Ext群的Gerstenhaber代数构造推广至带幺半中心系数的一般幺半范畴，显式刻画了双代数胚左模范畴情形下的结构并给出operad证明，推广了经典结果，同时提出更一般情形的猜想。

经典群在幺半群环上幺模行的初等作用

Elementary Action of Classical Groups on Unimodular Rows Over Monoid Rings

本文研究经典群在幺半群环幺模行上的初等作用传递性，给出辛群与正交群情形下的维数界，推导出经典群K₁的满稳定化界，推广了J. Gubeladze的线性群相关结果。