Bibliographical Items

The H×H-Theorem. If S is a compact connected monoid with group of units H and with E(S) = {0,1}, and if S/(H×H) (the space of orbits HsH) has a total order defning the quotient topology, then there is a one parameter semigroups I with E(I)=E(S) which commutes elementwise with H. (In particular the function (h, i)→hi∶H×I→HI=S is a surmorphism, and S is cylindrical.) This is Theorem VI in Elements of Compact Semigroups, by Hofmann and Mostert (p. 177). H. Carruth discovered a gap in the proof of this theorem in 1971. The methods of proof presented here differ from theose originally suggested and do not use peripherality. byt do use transformation group theory, and the authors' earlier results (Semigroup Forum 3 (1972), 31–42). The H×H-theorem is generalized to yield a theorem which belongs to the context of Theorem VIII in the Elements (p. 204):Theorem: Let S be a compact monoid such that the orbit space S/(H×H) is a totally ordered connected space with M(S) as its minimal point. If all regular D-classes are subsemigroups, then there is an I-semigroup with E(I)=E(S) which commutes elementwise with H. (In particular S=HI as in the H×H theorem). The sufficient condition about the regular D-classes is clearly necessary.). Further sample result:Theorem. IfH is a congruence in a compact connected monoid, with zero, then the centralizer of the group of units is connected.     

Primitive connected and additive theories of polygons

We study into monoids S the class of all S-polygons over which is primitive normal, primitive connected, or additive, that is, the monoids S the theory of any S-polygon over which is primitive normal, primitive connected, or additive. It is proved that the class of all S-polygons is primitive normal iff S is a linearly ordered monoid, and that it is primitive connected iff S is a group. It is pointed out that there exists no monoid S with an additive class of all S-polygons. __________ Translated from Algebra i Logika, Vol. 45, No. 3, pp. 300–313, May–June, 2006.     

A partial converse to a theorem of Tamari

Let M be a cancellative monoid such that the monoid ring ℤM has no zero divisors. We show that if the monoid consisting of all elements of ℤM with strictly positive coefficients has nonzero common right multiples, then M is left amenable.     

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.     

Finite groups with an almost regular automorphism of order four

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006.     

Approximate Amenability of Certain Semigroup Algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra ℓ¹(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then ℓ¹(S) is approximately amenable if and only if G is amenable. Moreover ℓ¹(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every M_a(S)-measurable subset B of S and s ∈ S the set sB is M_a(S)-measurable, it is proved that if the measure algebra M_a(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.     

On the second term in the spectral asymptotics of the maxwell operator in a filled resonator

It is shown that the characteristics of the Maxwell operator in a resonator with a smooth inhomogeneous anisotropic filler have constant multiplicity if and only if the matrices of dielectric permittivity and magnetic permeability are connected by the relation ε≡f μ, where f is a scalar-valued function. When ε≡f μ and the boundary is smooth and ideally conducting, the coefficient of λ² in the asymptotic expansion of the distribution function of the eigenvalues of the Maxwell operator turns out to be zero. When the multiplicity of the characteristics is variable, this coefficient can be either zero or nonzero. Bibliography: 22 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 48–79.     

A Note on Prehomomorphisms

We show that if G is a free group with basis X then any map θ from X to an inverse monoid S extends to a monoid prehomomorphism ψ: G\rightarrow S. As an application we give an affirmative answer to a problem of M. Petrich. 1980 Mathematics Subject Classification: Primary 20M10. September 14, 1999     

A counterexample in the cohomology of monoids

Let S be a commutative monoid, and 1et T be the maximal cancellative homomorphic image of S. The cohomology of S with coefficients in a left Z(S)-module D is given by Hⁿ(S,D)=Ext _Z(S) ⁿ (Z,D) as usual. [3] If from D we form the Z(T)-module D'=Hom_Z(S)(Z(T),D), then there is a natural homomorphism Hⁿ(T,D')→Hⁿ(S,D). If S is finite, then it follows from the results of [1] and [4] that this homomorphism is an isomorphism. The example presented below shows that this need no longer be an isomorphism if S is infinite. This research was supported in part by the National Science Foundation.     

Spherical orbits and representations of Uε(g)

Let U_ε(g) be the simply connected quantized enveloping algebra at roots of one associated to a finite dimensional complex simple Lie algebra g. The De Concini-Kac-Procesi conjecture on the dimension of the irreducible representations of U_ε(g) is proved for the representations corresponding to the spherical conjugacy classes of the simply connected algebraic group G with Lie algebra g. We achieve this result by means of a new characterization of the spherical conjugacy classes of G in terms of elements of the Weyl group.     

Value and perfection in stochastic games

A stochastic game isvalued if for every playerk there is a functionr ^k:S→R from the state spaceS to the real numbers such that for every ε>0 there is an ε equilibrium such that with probability at least 1−ε no states is reached where the future expected payoff for any playerk differs fromr ^k(s) by more than ε. We call a stochastic gamenormal if the state space is at most countable, there are finitely many players, at every state every player has only finitely many actions, and the payoffs are uniformly bounded and Borel measurable as functions on the histories of play. We demonstrate an example of a recursive two-person non-zero-sum normal stochastic game with only three non-absorbing states and limit average payoffs that is not valued (but does have ε equilibria for every positive ε). In this respect two-person non-zero-sum stochastic games are very different from their zero-sum varieties. N. Vieille proved that all such non-zero-sum games with finitely many states have an ε equilibrium for every positive ε, and our example shows that any proof of this result must be qualitatively different from the existence proofs for zero-sum games. To show that our example is not valued we need that the existence of ε equilibria for all positive ε implies a “perfection” property. Should there exist a normal stochastic game without an ε equilibrium for some ε>0, this perfection property may be useful for demonstrating this fact. Furthermore, our example sews some doubt concerning the existence of ε equilibria for two-person non-zero-sum recursive normal stochastic games with countably many states. This research was supported financially by the German Science Foundation (Deutsche Forschungsgemeinschaft) and the Center for High Performance Computing (Technical University, Dresden). The author thanks Ulrich Krengel and Heinrich Hering for their support of his habilitation at the University of Goettingen, of which this paper is a part.     

A study of operands in terms of maximal generalized orbits

A left operand X of a monoid S is called saturated if every generalized orbit (g.o.) in X is contained in a union of others. Every operand has a natural decomposition as a union of an operand admitting an irredundant cover by maximal g.o. ′s and of a saturated operand. There is a descending chain of suboperands of an operand, defined in terms of maximal g.o. ′s, which leads to the definition of the saturation length of an operand. S has no saturated operands if and only if S satisfies the a.c.c. on orbits. Full proofs will be given in [2]. Communicated by A. H. Clifford     

关于所有强平坦右S-系是正则系的幺半群(英)

本论文考虑了所有强平坦右S-系是正则系的幺半群的刻画,证明了所有强平坦右S-系是正则S-系当且仅当S是右PSF幺半群并且S的每一个左coilpasible子幺半群包含左零元．该结果对Kilp和Knauer在文献[7]中的问题给出了一个新的回答．     

Left zero covering homomorphisms of laminated nearrings

Nearrings here are right nearrings. LetN be a nearring and fix an element α εN. Form another nearring N_α by taking addition to be the same as the addition inN but define the productx ⊗y of two elementsx, y ε N_α byx ⊗y =xay. The nearring N_α is referred to as a laminated nearring ofN andN is referred to as the base nearring. The element α is called the laminating element or the laminator. An elementx of a nearingN is a left zero ifxy =x for ally εN. A homomorphismϕ from a nearringN ₁ into a nearringN ₂ is a left zero covering homomorphism if for each left zeroy εN ₂,ϕ(x) =y for somex εN ₁. The left zero covering homomorphisms from one laminated nearring into another are investigated where the base nearring is the nearring of all continuous selfmaps of the Euclidean group ℝ² under pointwise addition and composition and the laminators are complex polynomials. Finally, it is shown that one can determine whether or not two such laminated nearrings are isomorphic simply by inspecting the coefficients of the two laminating polynomials.     

具有阿贝尔Sylow 2-子群的有限群的整群环的正规化子性质

Mazur猜想:具有阿贝尔Sylow 2-子群的有限群有正规化子性质.设G是一个有限群,N是G的一个正规子群且Z(G/N)仅有平凡单位,本文建立了由Z(G/N)中单位诱导的G的自同构与N的Coleman自同构之间的联系,在此基础上证明了若G是一个具有阿贝尔Sylow 2-子群的有限群且Z(G/F*(G))仅有平凡单位,则Mazur猜想对G成立.     

广义逆多项式模的co-Hopf性质

设R是结合环(可以没有单位元),(S,≤)是严格全序幺半群,序≤是Artin的且对任意s∈S,有0≤s,则对任意具有性质(F)的左R-模M,[MS,≤]是co-Hopf左[[RS,≤]]一模当且仅当M是co-Hopf左R-模.     

A poisson formula for solvable Lie groups

Given a probability measure μ on a locally compact second countable groupG the space of bounded μ-harmonic functions can be identified withL ^∞(η, α) where (η, α) is a BorelG-space with a σ-finite quasiinvariant measure α. Our goal is to show that when μ is an arbitrary spread out probability measure on a connected solvable Lie groupG then the μ-boundary (η, α) is a contractive homogeneous space ofG. Our approach is based on a study of a class of strongly approximately transitive (SAT) actions ofG. A BorelG-space η with a σ-finite quasiinvariant measure α is called SAT if it admits a probability measurev≪α, such that for every Borel set A with α(A)≠0 and every ε>0 there existsg∈G with ν(gA)>1−ε. Every μ-boundary is a standard SATG-space. We show that for a connected solvable Lie group every standard SATG-space is transitive, characterize subgroupsH⊆G such that the homogeneous spaceG/H is SAT, and establish that the following conditions are equivalent forG/H: (a)G/H is SAT; (b)G/H is contractive; (c)G/H is an equivariant image of a μ-boundary.     

Sur les automates qui reconnaissent une famille de langages

The main results about automatas and the languages they accept are easily extended to automatas which recognize a family of languages (L_i)_iεI of a free monoid, that is to automatas which recognize simultaneously all the languages L_i. This generalization enhances the notion of automata of type (p,r) introduced by S. Eilenberg [4]. In a similar way the notion of syntactic monoid of a family of languages extends the notion of syntactic monoid of a language. This extension is far from being trivial since we show that every finite monoid is the syntactic monoid of a recognizable partition of a free monoid, though this is false for the syntactic monoids of languages.      

TC semigroups and inflations

An algebraA satisfiesTC (the term condition) if for any and anyn + 1-ary termp.TC algebras have been extensively studied. We previously determined the structure of allTC semigroups. We use this result to show that ifS is aTC semigroup thenS _E = {a ε S | ax is an idempotent for somex ε S} is an inflation ofS _Reg (the set of regular elements ofS) andS _Reg ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup. As a corollary of this result, we show thatS is a semisimpleTC semigroup iffS ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup.