Casey Donoven
In this article, we show that a group $G$ is the union of two proper subsemigroups if and only if $G$ has a nontrivial left-orderable quotient. Furthermore, if $G$ is the union of two proper semigroups, then there exists a minimum normal subgroup $N\unlhd G$ for which $G/N$ is left-orderable and nontrivial.
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Our aim is to find some new links between linear (circular) orderability of groups and topological dynamics. We suggest natural analogs of the concept of algebraic orderability for topological groups involving order-preserving actions on compact spaces and the corresponding enveloping semigroups in the sense of R. Ellis. This approach leads to several natural questions. Some of them might be useful also for discrete (countable) orderable groups.
Let $\mathbb{A} = (A, \cdot)$ be a semigroup. We generalize some recent results by G. A. Freiman, M. Herzog and coauthors on the structure theory of set addition from the context of linearly orderable groups to linearly orderable semigroups, where we say that $\mathbb{A}$ is linearly orderable if there exists a total order $\le$ on $A$ such that $xz < yz$ and $zx < zy$ for all $x,y,z \in A$ with $x < y$. In particular, we find that if $S$ is a finite subset of $A$ generating a non-abelian subsemigroup of $\mathbb{A}$, then $|S^2| \ge 3|S|-2$. On the road to this goal, we also prove a number of…
We show that every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) has linear growth. This implies that the the corresponding semigroup algebra is a PI algebra.
Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.
Every semigroup which is a finite disjoint union of copies of the free mono- genic semigroup (natural numbers under addition) has soluble word prob- lem and soluble membership problem. Efficient algorithms are given for both problems.
We prove that, up to isomorphism and anti-isomorphism, there are only two semigroups which are the union of two copies of the free monogenic semigroup. Similarly, there are only nine semigroups which are the union of three copies of the free monogenic semigroup. We provide finite presentations for each of these semigroups.
For a semigroup $S$, the covering number of $S$ with respect to semigroups, $σ_s(S)$, is the minimum number of proper subsemigroups of $S$ whose union is $S$. This article investigates covering numbers of semigroups and analogously defined covering numbers of inverse semigroups and monoids. Our three main theorems give a complete description of the covering number of finite semigroups, finite inverse semigroups, and monoids (modulo groups and infinite semigroups). For a finite semigroup that is neither monogenic nor a group, its covering number is two. For all $n\geq 2$, there exists an invers…
A semigroup $S$ is an equational domain if any finite union of algebraic sets over $S$ is algebraic. We prove that every nontrivial semigroup in the standard language $\{\cdot\}$ is not an equational domain.
A semigroup $S$ is an equational domain if any finite union of algebraic sets over $S$ is algebraic. We prove that if an inverse semigroup $S$ is an equational domain in the extended language $\{\cdot,{}^{-1}\}\cup\{s|s\in S\}$ then $S$ is a group.
Semiring arising as Lattice of Groupsemirings
本文研究由分配格标引的群半环类构造半环，证明若群半环构成强分配格，则所得半环的乘法半群为Clifford半群、加法半群为左正规带，且所有组成群彼此同构。
In this paper we continue the investigations on the algebraic structure of a finite semigroup $S$ that is determined by its associated upper non-nilpotent graph $\mathcal{N}_{S}$. The vertices of this graph are the elements of $S$ and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). We introduce a class of semigroups in which the Mal'cev nilpotent property lifts through ideal chains. We call this the class of \B\ semigroups. The definition is such that the global information that a semigroup is not nilpotent induces local information, i.e. …
There has been interest recently concerning when a left ordered group is locally indicable. Bergman and Tararin have shown that not all left ordered groups are locally indicable, but all known examples contain a nonabelian free subgroup. We shall show for a large class of groups not containing a nonabelian free subgroup, that any left ordered group in this class is locally indicable. Specifically this class is the smallest class of groups containing NS and Thompson's group of piecewise linear homeomorphisms of the unit interval, and is closed under taking subgroups, quotient groups, extensions…
Let $A$ be a finite dimensional $Q-$algebra and $Γsubset A$ a $Z-$order. We classify those $A$ with the property that $Z^2$ does not embed in $\mathcal{U}(Γ)$. We call this last property the hyperbolic property. We apply this in the case that $A = KS$ a semigroup algebra with $K = Q$ or $K = Q(\sqrt{-d})$. In particular, when $KS$ is semi-simple and has no nilpotent elements, we prove that $S$ is an inverse semigroup which is the disjoint union of Higman groups and at most one cyclic group $C_n$ with $n \in \{5,8,12\}$.
We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented \[G(m,n,k) = \left\langle {a,b;{a^m} = 1,{b^n} = 1,{a^b} = {a^k}} \right\rangle \quad (m,n,k\in\mathbb{Z}^+)\] with $(m,k - 1) = 1$ and $n = in{d_m}(k),$ the smallest positive integer for which ${k^n} = 1\,\pmod m,$ with the conjugate of $a$ by $b$ written ${a^b}( = {b^{ - 1}}ab).$ The \emph{right} and \emph{left commutation semigroups of} $G,$ denoted ${\rm P}(G)$ and $Λ(G),$ are the semigroups of mappings generated by $ρ(g):G \to G$ and $λ(g):G \to G$ defined by $(x)ρ(g) = …
Set-theoretical solutions of the pentagon equation on Clifford semigroups
本文研究 Clifford 半群上五边形方程的集合论解，给出了幂等不变解的完整描述，并从每个群上的解出发构造了一族幂等不动解。
Residual coherentization of balanced residuated partially ordered semigroups
研究平衡剩余偏序半群，通过构造剩余相干化$\mathcal C_{\mathrm r}(\mathbf M)$，给出规范分解 - 重构理论，其块定义规范分量，商骨架记录目标分量，还与子半格稳定可见性分解比较，显示前者划分更精细。
Submonoids of Infinite Symmetric Inverse Monoids
本论文研究无限集上对称逆幺半群的子半群结构，通过最大子半群分类、拓扑闭子半群刻画及Bergman-Shelah预序分类三种方法，并提出了相关猜想。
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