On a class of finite solvable groups

A finite solvable group G is called an X-group if the subnormal subgroups of G permute with all the system normalizers of G. It is our purpose here to determine some of the properties of X-groups. Subgroups and quotient groups of X-groups are X-groups. Let M and N be normal subgroups of a group G of relatively prime order. If G/M and G/N are X-groups, then G is also an X-group. Let the nilpotent residual L of G be abelian. Then G is an X-group if and only if G acts by conjugation on L as a group of power automorphisms.     

Trivial nuclear ideals of alternative algebras

Let A be a free alternative Φ-algebra, where Φ is an associative commutative ring with 1, containing 1/6, and g(y, z, t, v, x, x)=2[J_({[y, z], t, x}_, x, v)+J_({[y,x], z, x}_, t, v)], where [x, y]=xy−yx, J_(x, y, z)=[[x, y], z]+[[z, x], y]+[[y, z], x], {x, y, z}_=J_(x, y, z)+3[x, [y, z]]. We construct trivial nuclear ideals of A, that is, nonzero ideals with zero multiplication, lying in the associative center of A. In particular, it is shown that if G and B are fully invariant ideals of A on k≥7 free generators, generated by a function g and by double commutators, respectively, then GB+BG is a nuclear ideal of A. This implies that an unmized alternative algebra satisfies GB=BG=0. If an unmixed algebra is finitely generated, then G=0. In addition, we prove that if R is an unmixed solvable alternative algebra then (R^N)²=0 for some N. Translated fromAlgebra i Logika, Vol. 36, No. 1, pp. 97–115, January–February, 1997.     

A fully invariant ideal of alternative algebras

Let ϕ be an associative commutative ring with 1, containing 1/6, and A be an alternative ϕ-algebra. Let D be an associator ideal of A and H a fully invariant ideal of A, generated by all elements of the form h(y, z, t, x, x)=[{[y, z], t, x}-, x]+[{[y, x], z, x}-, t], where [x, y]=xy−yx, {x, y, z}-=[[x, y], z]−[[x, z], y]+2[x,[y, z]]. Here we consider an ideal Q=H∩D and prove that Q⁴=0 in the algebra A. If A is unmixed, then HD=0, DH=0, and Q²=0 in particular. If A is a finitely generated unmixed algebra, then the ideal H lies in its associative center and Q=0. It follows that any finitely generated purely alternative algebra satisfies the identity h(y,z,t,x,x)=0. We also show that a fully invariant ideal H₀ of the unmixed algebra A, generated by all elements of the form h(x, z, t, x, x), lies in its associative center and H₀∩D=0. Consequently, every purely alternative algebra satisfies the identity h(x,z,t,x,x)=0. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 323–340, May–June, 1997.     

Free and almost-free subsemigroups of a free semigroup

A subsemigroup S of a free semigroup F(Σ) is almost-free if there is a free subsemigroupT such that S?T?F(Σ) and T/S is finite. It is shown that it is decidable whether a subsemigroup generated by a regular subset of F(Σ) is almost-free. Sufficient- conditions are given such that if a family F of subsets of F(Σ) satisfies these conditions, then it is undecidable for L∈F whether the subsemigroup generated by L is free and also whether it is almost-free.     

The group PSL(2,q) is not the multiplication group of a loop

ABSTRACT

We introduce the notion of weak transitivity for torsion-free abelian groups. A torsion-free abelian group G is called weakly transitive if for any pair of elements x, y ∈ G and endomorphisms ?, ψ ∈ End(G) such that x? = y, yψ = x, there exists an automorphism of G mapping x onto y. It is shown that every suitable ring can be realized as the endomorphism ring of a weakly transitive torsion-free abelian group, and we characterize up to a number-theoretical property the separable weakly transitive torsion-free abelian groups.     

Regularity and Green’s relations on semigroups of transformations with restricted range that preserve an equivalence

Let Y be a subset of X and T(X, Y) the set of all functions from X into Y. Then, under the operation of composition, T(X, Y) is a subsemigroup of the full transformation semigroup T(X). Let E be an equivalence on X. Define a subsemigroup $$T_E(X,Y)$$ of T(X, Y) by $$\begin{aligned} T_E(X,Y)=\{\alpha \in T(X,Y):\forall (x,y)\in E, (x\alpha ,y\alpha )\in E\}. \end{aligned}$$Then $$T_E(X,Y)$$ is the semigroup of all continuous self-maps of the topological space X for which all E-classes form a basis carrying X into a subspace Y. In this paper, we give a necessary and sufficient condition for $$T_E(X,Y)$$ to be regular and characterize Green’s relations on $$T_E(X,Y)$$. Our work extends previous results found in the literature.     

Tamely uniform orders

The following two results are proven.

(i) Let G be a finitely generated torsion-free linear group. If every torsion-free section of G is an R-group, then G is soluble of finite rank. Conversely, if G has finite rank, then it has a subgroup of finite index, in which every torsion-free section is an R-group.

Let G be a finitely generated torsion-free soluble group. If in every torsion-free section of G the normalizer of each isolated subgroup is isolated, then G has finite rank. Conversely, if G has finite rank, then it has a subgroup K of finite index such that in every torsion-free section of K the normalizer of each isolated subgroup is isolated.     

具有Q-正则*-断面的正则半群上的*-同余格

文中给出了一个具有正则*-断面正则半群的例子,该半群同时存在非平凡*-同余和非平凡的非*-同余;证明了正则*-断面上的每个*-同余都能扩张成整个半群上的*-同余;刻划了*-同余和*-同余格;定义了*-同余格上的两个完全同余T*FS和T*S*;研究了*-同余格上的完全同余T*S*, T*, T*l, Tr, U*和V*, 给出了这些同余的类中的极值同余(除U*, V*外).     

Sur la (-k)-demi-reconstructibilité des tournois finis

Given a tournament T, we define the dual T^* of T by T^*(x,y) = T(y,x). A tournament T′ is hemimorphic to T if it is isomorphic to T or T^*. A tournament defined on n elements is (-k-reconstructible (resp. (-k)-half-reconstructible) if it is determined up to isomorphism (resp. up to hemimorphism), by its restrictions to subsets of (n - k) elements. From [2] follows the (-k)-half-reconstructibility of finite tournaments (with n ≥ (7 + k) elements), for all k > 7. In this Note, we establish the (-k)-half-reconstructibility of finite tournaments (with n ≥ (12 + k) elements), for all k4,5,6. We then connect the problems of the (-3)- and the (−2)-half-reconstruction of these tournaments to two problems (yet open) of reconstruction. Finally, by using counterexamples of P.K. Stockmeyer [14], we show that, generally, the finite tournaments are not (-k)-half-reconstructible.     

Hamiltonian loops from the ergodic point of view

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S¹→G is called strictly ergodic if for some irrational number α the associated skew product map T:S¹×Y→S¹×Y defined by T(t,y)=(t+α,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on the homotopy classes of smooth strictly ergodic loops in the framework of Hofer’s bi-invariant geometry on G. Namely, we prove that their asymptotic Hofer’s norm must vanish. This result provides a link between ergodic theory and symplectic topology. Received July 7, 1998 / final version received September 14, 1998     

Automatic subsemigroups of free products

We consider the automaticity of subsemigroups of free products of semigroups, proving that subsemigroups of free products, with all generators having length greater than one in the free product, are automatic. As a corollary, we show that if S is a free product of semigroups that are either finite or free, then any finitely generated subsemigroup of S is automatic. In particular, any finitely generated subsemigroup of a free product of finite or monogenic semigroups is automatic.     

The inverse hull of the free semigroup on a set X

In this paper the inverse hull of the free semigroup on a non-empty set X is characterized. It is shown that if F^O is a certain semilattice and (G,*) is the free group on X, then is isomorphic to a Rees factor semigroup of a subsemigroup of (F^O x G, #), where (A,w)#(B,u) = ((A*w B)*w^–1, w*u). Several properties of , including Green's relations on are also determined.     

The quasivariety generated by a torsion-free Abelian-by-finite group

Let L_q(qG) be the quasivariety lattice contained in a quasivariety generated by a group G. It is proved that if G is a finitely generated torsion-free group in (i.e., G is an extension of an Abelian group by a group of exponent 2ⁿ), which is a split extension of an Abelian group by a cyclic group, then the lattice L_q(qG) is a finite chain. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 407–427, July–August, 2007.     

保持序和等价关系的自然偏序变换半群

设X为一个集合,■_X为X上的全变换半群.设E是X上的一个等价关系,定义T_E(X)={f∈■_X:■(x,y)∈E,(f(x),f(y))∈E},则T_E(X)是由等价关系E所确定的■_X的子半群.本文中,所考虑的集合X是一个有限全序集,同时E是非平凡的且所有的E-类都是凸集.显然■_E(X)={f∈T_E(X):■_x,y∈X,x≤y蕴涵f(x)≤f(y)}是T_E(X)的一个子半群.我们赋予■_E(X)自然偏序并讨论何时■_E(X)中的两个元素是关于这个偏序是相关的,然后确定■_E(X)中那些关于≤是相容的元素.此外,还描述了极大(极小)元和覆盖元.     

A note on intersections of free submonoids of a free monoid

According to a theorem of Tilson [6] any intersection of free submonoids of a free monoid is free. Here we consider intersections of the form {x, y}^* ∩ {u, v}^*, where x, y, u and v are words in a finitely generated free monoid Σ^*, and show that if both the monoids {x, y}^* and {u, v}^* are of the rank two, then the intersection is a free monoid generated either by (at most) two words or by a regular language of the form β₀ + β((γ(1+ δ + ... δ^t))^*ε for some words β⁰, β, γ, δ and ε, and some integer t≥0. An example is given showing that the latter possibility may occur for each t≥0 with nonempty values of the words.     

Green's Equivalences on Semigroups of Transformations Preserving Order and an Equivalence Relation

Let ${\cal T}_X$ be the full transformation semigroup on the set $X$, \[ T_{E}(X)=\{f\in {\cal T}_X\colon \ \forall(a,b)\in E,(f(a),f(b))\in E\} \] be the subsemigroup of ${\cal T}_X$ determined by an equivalence $E$ on $X$. In this paper the set $X$ under consideration is a totally ordered set with $mn$ points where $m\geq 2$ and $n\geq 3$. The equivalence $E$ has $m$ classes each of which contains $n$ consecutive points. The set of all order preserving transformations in $T_{E}(X)$ forms a subsemigroup of $T_E(X)$ denoted by \[ {\cal O}_{E}(X)=\{f\in T_{E}(X)\colon \ \forall\, x, y\in X, \ x\leq y \mbox{ implies } f(x)\leq f(y)\}. \] The nature of regular elements in ${\cal O}_{E}(X)$ is described and the Green's equivalences on ${\cal O}_{E}(X)$ are characterized completely.     

Minimal non-group twisted subsets containing involutions

A subset K of a group G is said to be twisted if 1 ∈ K and xy⁻¹x ∈ K for any x, y ∈ K. We explore finite twisted subsets with involutions which are themselves not subgroups but every proper twisted subset of which is. Groups that are generated by such twisted subsets are classified. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 459–482, July–August, 2007.     

Non-overlapping control systems on Lie groups

LetG be a Lie group with Lie algebraL(G) and let Ω be a non-empty subset ofL(G). If Ω is interpreted as the set of controls, then the set of elements attainable from the identity for the system Ω is a subsemigroup ofG. A system Ω is called anon-overlapping control system if any element attainable for Ω is only attainable at one time. In this paper, we show that a compact convex generating nonoverlapping control systems on a connected Lie group must be contained inX+E for someX∈L(G)\E, where E is a subspace of codimension one containing the commutator, and the homomorphism from the attainable semigroup intoR ⁺ extends continuously to the whole group in the case of solvable Lie groups. This work is done under the support of TGRC-KOSEF.     

一类特殊的Koszul Calabi-Yau DG代数

假设一个连通上链DG代数A的基分次代数A~#或者同调分次代数H(A)是由一次元素x,y生成的代数kx,y/(xy+yx).本文证明A是Koszul Calabi-Yau DG代数.