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The purpose of this paper is to connect the notion of Gorenstein dimension with AS-Gorenstein algebras. In particular, we show that a noetherian connected graded algebra having a balanced dualizing complex is AS-Gorenstein if the balanced dualizing complex has finite Gorenstein dimension. As a preparation, we generalize the Auslander–Bridger formula to the class of noncommutative noetherian connected graded algebras having balanced dualizing complexes. [ABSTRACT FROM AUTHOR]

Letw(x,y) be a word in two variables and 𝔚 the variety determined byw. In this paper we raise the following question: if for every pair of elementsa,bin a groupGthere existsg ∈ Gsuch thatw(ag,b) = 1, under what conditions does the groupGbelong to 𝔚? In particular, we consider then-Engel wordw(x,y) = [x,ny]. We show that in this case the property is satisfied when the groupGis metabelian. Ifn = 2, then we extend this result to the class of all solvable groups. [ABSTRACT FROM AUTHOR]

Among other results of this paper, we single out the following ones. We show that division rings are the only division semirings over which the categories of semimodules are Schreier varieties, i.e., all subsemimodules of free semimodules are free too. We a complete description of division semiringsRover which the categories of semimodules ℳRarep-Schreier varieties, i.e., varieties whose all projective algebras are free. We give a complete description of proper division semiringsRwhose categories of semimodules ℳR(X)over the polynomial semiringsR(X) overR, in not necessary commuting variablesX, arep-Schreier varieties. We show that the categories of semimodules ℳR(X)over the polynomial semiringsR(X) overN-valued semiringsR, in particular ℳN(X), arep-Schreier varieties. We also show that forN-valued semiringsS, the semimodule categories ℳSnever are Schreier varieties. [ABSTRACT FROM PUBLISHER]

Using the Tate–Poitou duality, Sansuc proved that the group III1(K,G) is stablyK-birational invariant ofGfor a connected linear algebraic group defined over a number fieldK. In this paper, we consider the case whereKis a function field, in one variable over a PAC field (Kis a “good” field in sense Colliot–Thélène and Kunyavskii). We show that the group III1(K,G) is stablyK-birational invariant whenGis a connected reductiveK-group. Since we no longer have the Tate–Poitou duality at our disposition, we use the flasque resolutions of Colliot–Thélène and Sansuc. En utilisant la dualité de Tate–Poitou, Sansuc a établi le caractère d'invariant stablementK-birationnel du groupe III1(K,G) pour un groupe algébrique linéaire connexeGdéfini sur un corps de nombresK. Dans cet article, nous considérons le cas oùKest un corps de fonctions en une variable sur un corps PAC (Kest un “bon” corps au sens de Colliot–Thélène et Kunyavskii). Nous montrons le caractère d'invariant stablementK-birationnel du groupe III1(K,G) pour unK-groupe réductif connexeG. Comme nous n'avons plus à notre disposition la dualité de Tate–Poitou, nous devons utiliser les résolutions flasques de Colliot–Thélène et Sansuc. [ABSTRACT FROM PUBLISHER]

The purpose of this paper is the study of hypergroups associated with hypergraphs. In this regards, we construct a ρ-hypergroup by means a given hypergraph by defining a special relation ρ, and then we investigate some related properties. Further, we introduce a special product of ρ-hypergroups. Also, we bridge between subhypergraphs and subhypergroups. Finally, the fundamental relation of a ρ-hypergroup is studied. [ABSTRACT FROM AUTHOR]

Finite groupsGsuch thatG/Z(G) ≈C2 × C2whereC2denotes a cyclic group of order 2 andZ(G) is the center ofGwere studied in [5] and were used to classify finite loops with alternative loop algebras. In this paper we extend this result to finitely generated groups such thatG/Z(G) ≈Cp × CpwhereCpdenotes a cyclic group of prime orderpand provide an explicit description of all such groups. [ABSTRACT FROM AUTHOR]

In the paper we describe the subcategory of the category of ℤ-graded Lie algebras which is equivalent to the category of Jordan pairs via a functorial modification of the Tits, Kantor, and Koecher TKK construction. For instance, we prove thatL = L−1⊕L0⊕L1can be constructed from a Jordan pair if and only ifL0 = [L−1,L1] and the second graded homology groupis trivial. Similar descriptions are obtained for Jordan triple systems and Jordan algebras. New functorial versions of the TKK construction are given for pairs and algebras. [ABSTRACT FROM AUTHOR]

LetGbe a group. By using a family 𝒜 of subsets of automorphisms ofG, we introduced a simple graph Γ𝒜(G), which is a generalization of the non-commuting graph. In this paper, we study the combinatorial properties of Γ𝒜(G). [ABSTRACT FROM AUTHOR]

A subgroupHof a finite groupGis calledc*-normal inGif there exists a normal subgroupKofGsuch thatG = HKandH∩KisS-quasinormally embedded inG. In this paper, the structure of a finite groupGwith somec*-normal subgroups of Sylowp-subgroups with an any fixed order is characterized and some known related results onp-nilpotency of finite groups are generalized. [ABSTRACT FROM AUTHOR]

Abelian groups whose endomorphism rings are von Neumann regular have been extensively investigated in the literature. In this paper, we study modules whose endomorphism rings are von Neumann regular, which we call endoregular modules. We provide characterizations of endoregular modules and investigate their properties. Some classes of ringsRare characterized in terms of endoregularR-modules. It is shown that a direct summand of an endoregular module inherits the property, while a direct sum of endoregular modules does not. Necessary and sufficient conditions for a finite direct sum of endoregular modules to be an endoregular module are provided. As a special case, modules whose endomorphism rings are semisimple artinian are characterized. We provide a precise description of an indecomposable endoregular module over an arbitrary commutative ring. A structure theorem for extending an endoregular abelian group is also provided. [ABSTRACT FROM PUBLISHER]

In this paper we characterize the paramedial, comedial, and coparamedial pairs of quasigroup operations and paramedial, comedial, and coparamedial algebras with the quasigroup operations. Then we characterize paramedial, comedial, and coparamedial algebras with a unit element and an idempotent regular element. [ABSTRACT FROM PUBLISHER]

A power automorphism θ of a group G is said to be pre-fixed-point-free if CG(θ) is an elementary abelian 2-group. G is called an E-group if G has a pre-fixed-point-free power automorphism. In this paper, finite E-groups, together with all their pre-fixed-point-free power automorphisms, are completely determined. Moreover, a characteristic of finite abelian groups is given, which explains some known facts concerning power automorphisms. [ABSTRACT FROM PUBLISHER]

The notion of cohomological periodicity after 1-step has been studied by Talelli in [7, 8], and [9]. If a group G has periodic cohomology after 1-step, then G is the fundamental group of a graph of finite groups, which have periodic cohomology of the same period. Also, the fundamental group of a tree of finite groups, which have periodic cohomology of the same period, has periodic cohomology after 1-step. In this paper, we show that if a group G has only cyclic finite subgroups and is the fundamental group of a certain tree of groups, which have -steps. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide a detailed structure description of derived subpolygroups of polygroups. We introduce the concept of perfect and solvable polygroups and we give some results in this respect. Finally, we discuss on τ-multi-semi-direct hyperproduct of polygroups. [ABSTRACT FROM AUTHOR]

In [1], the authors proved that the maximum sum of element orders on finite groups of the same order occurs in cyclic group. In this paper we obtain the structure of groups having maximum sum of element orders on noncyclic nilpotent group of the same order. [ABSTRACT FROM AUTHOR]

A subgroupHofGis called ℳp-supplemented inGif there exists a subgroupBofGsuch thatG = HBandTB < Gfor every maximal subgroupTofHwith |H:T| =pα. In this paper, we investigate the influence of ℳp-supplemented subgroup and some conditions forp-nilpotency andp-supersolvability of finite groups are obtained. [ABSTRACT FROM AUTHOR]

The notion of almost left factorizability and the results on almost left factorizable weakly ample semigroups, due to Gomes and the author, are adapted for restriction semigroups. The main result of the paper is that each restriction semigroup is embeddable into an almost left factorizable restriction semigroup. This generalizes a fundamental result of the structure theory of inverse semigroups. [ABSTRACT FROM AUTHOR]

An element of a group is called an iicentral if the conjugucy class of that element is equal to the coset of the commutator subgroup containing that element. A group is called Camilla group if every element outside the commutator subgroup is anticentral. In this paper, we investigate the structure of locally finite groups with an anticentral element. Moreover, we construct some non-periodic examples of Camilla groups, which are not locally solvable. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions for the existence and uniqueness of normal forms of sequences of HNN extensions defined by Bokut'. Furthermore, we show that under an assumption, which holds for various applications, such normal forms always exist (but might not be unique). The conditions are amenable to be used in automatic theorem provers. We discuss also how to obtain a Grobner-Shirshov basis from the rewrite rules of Bokut' normal forms under certain assumptions. Finally, we provide an application drawn from a paper of Aanderaa and Cohen to illustrate the sufficiency conditions. [ABSTRACT FROM AUTHOR]

Hanna Neumann asked whether it was possible for two non-isomorphic residually nilpotent finitely generated (fg) groups, one of them free, to share the lower central sequence. Baumslag answered the question in the affirmative and thus gave rise to parafree groups. A group G is termed parafree of rank n if it is residually nilpotent and shares the same lower central sequence with a free group of rank n. The deviation of a fg parafree group G of rank n is the difference μ(G) - n, where μ(G) is the minimum possible number of generators of G. Let G be fg; then Hom(G, SL2) inherits the structure of an algebraic variety, denoted by R(G). If G is an n generated parafree group, then the deviation of G is 0 iff Dim(R(G)) = 3n. It is known that for n ≥ 2 there exist infinitely many parafree groups of rank n and deviation 1 with non-isomorphic representation varieties of dimension 3n. In this paper it is shown that given integers n ≥ 2 and k ≥ 1, there exists infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, there exist infinitely many parafree groups G of rank n with Dim(R(G)) > q, where q ≥ 3n is an arbitrary integer. [ABSTRACT FROM AUTHOR]

In this paper we introduce a construction called the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S which extends Ribenboim's construction of generalized power series rings. In the case when S is totally ordered or commutative aperiodic, and ω(s) is constant on idempotents for some s ∈ S∖{1}, we give sufficient and necessary conditions on R and S such that the ring R[[S, ω]] is von Neumann regular, and we show that the von Neumann regularity of the ring R[[S, ω]] is equivalent to its semisimplicity. We also give a characterization of the strong regularity of the ring R[[S, ω]]. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. An algebraic element α over R is called an “integral algebraic element” over R if there exists a minimal polynomial of α over R which is monic. Let R be a direct product of m indecomposable rings Ri, m ∈ ℕ. Denote by RG the group ring of G over R and by R* the multiplicative group of R.Let G be a finite Abelian group of exponent n and n ∈ R. In this paper we give a decomposition of RG, up to isomorphism, into a direct sum of extensions of the ring R, taking into account the number of the repetitions of these extensions. If the ring R is a field, then this result is proved in Perlis and Walker (1950).Let G be a splitting Abelian group with a torsion subgroup G0. Denote by Gp the p -component of G. We give a description of the unit group U(RG) of RG in the following cases: when Ri is a ring of prime characteristic pi, G0/Gpi is finite and the exponent of G0/Gpi belongs to Ri*;when Ri is of characteristic zero, Ri has no nilpotents, G0 is finite of exponent n and n ∈ Ri*.For the establishment of these results we prove that if the ring R is indecomposable and n ∈ R*, then: the cyclotomic polynomial Φn(x) has a unique decomposition in a product of monic irreducible factors over R; andif α and β are integral algebraic elements over R which are roots of monic irreducible divisors of the cyclotomic polynomial Φn(x), then the rings R[α] and R[β] are isomorphic. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to construct the character tables of the parabolic subgroups of the Chevalley groups G 2 ( q ), where q is a power of a prime p > 3. [ABSTRACT FROM AUTHOR]

In this paper we classify, up to exact isomorphism, the algebras in the title which contain a faithful element and in which every non-trivial table subset has dimension at least five. [ABSTRACT FROM AUTHOR]

A subgroup K of a finite group G is called an ℋ-subgroup of G if the following condition is satisfied: The set of all ℋ-subgroups of a finite group G will be denoted by ℋ( G ). In this paper, we investigate the structure of a finite group G under the assumption that certain subgroups of prime power orders belong to ℋ( G ). Communicated by M. Dixon [ABSTRACT FROM AUTHOR]

Let G be a group and let N be a normal subgroup of G. We set cd( G | N ) to be the degrees of the irreducible characters of G whose kernels do not contain N. We associate a graph with this set. The vertices of this graph are the primes dividing degrees in cd( G | N ), and there is an edge between p and q if pq divides some degree in cd( G | N ). In this paper, we study this graph when it is disconnected, and we study its diameter when it is connected. [ABSTRACT FROM AUTHOR]

If X and Y are sets, we let P ( X , Y ) denote the set of all partial transformations from X into Y (that is, all mappings whose domain and range are subsets of X and Y , respectively). We define an operation * on P ( X , Y ) by choosing θ &Element; P ( Y , X ) and writing: α*β = α °θ°β, for each α, β &Element; P ( X , Y ). Then ( P ( X , Y ), *) is a semigroup, and some authors have determined when this is regular (Magill and Subbiah, 1975), when it contains a “proper dense subsemigroup” (Wasanawichit and Kemprasit, 2002) and when it is factorisable (Saengsura, 2001). In this paper, we extend the latter work to certain subsemigroups of ( P ( X , Y ), *). We also consider the corresponding idea for partial linear transformations from one vector space into another. In this way, we generalise known results for total transformations and for injective partial transformations between sets, and we establish new results for linear transformations between vector spaces. [ABSTRACT FROM AUTHOR]

In this paper we deduce an upper bound of the periods of two-dimensional periodic arrays over a Galois ring and give a root characterization of maximal periodic arrays. [ABSTRACT FROM AUTHOR]

H is called a G -subgroup of a hyperbolic group G if for any finite subset M &NotSubset; G there exists a homomorphism from G onto a non-elementary hyperbolic group G 1 that is surjective on H and injective on M. In his paper in 1993 A. Ol'shanskii gave a description of all G-subgroups in any given non-elementary hyperbolic group G. Here we show that for the same class of G-subgroups the finiteness assumption on M (under certain natural conditions) can be replaced by an assumption of quasiconvexity. [ABSTRACT FROM AUTHOR]

A polynomial form f, is a not necessarily linear map, from an infinite module over a ring &zfr; to a finite abelian group of exponent n satisfying some additional conditions. Denote the zeros of f by Ω f . We show it satisfies a weak closure condition. Among all &zfr;-submodules of finite index, there is a submodule B such that |f (B)| (the order of the subset f (B)) is as small as possible. f (B) is called the final value of f and D. S. Passman asks if f (B) is necessarily a subgroup of S. This paper shows that if the degree of f ≤ 2 then the final value is a subgroup and if the form f has arbitrary degree from an finitely generated infinite abelian group, then the final value is 0. [ABSTRACT FROM AUTHOR]

The test rank tr( G ) of G is the minimum cardinality of a test set. In this paper we prove: I. Let N = N rc be a free nilpotent group of rank r = 2 and class c = 2. Then (i) tr( N ) = 2 for r odd and c = 2; (ii) tr( N ) = 1 in all other cases; (iii) an element g ? N 2 q , 2 is a test element if and only if it can be written as g = · s , where t 1 ,..., t q are nonzero integers. II. Let F r be a free group of rank r = 2 and A n be a free abelian group of rank n = 1. Then the group G = G rn = F r × A n has the test rank n . [ABSTRACT FROM AUTHOR]

In this paper we prove that a finite group G is isomorphic to the finite simple group L n ( q ) with n = 3 if and only if they have the same set of order of solvable subgroups. [ABSTRACT FROM AUTHOR]

In the paper, a Frattini-like subgroup associated with a class ?? of simple groups is introduced and analyzed. The corresponding ??-saturated formations are exactly the ??-local ones introduced by Förster. Our techniques are also useful to highlight the properties and behavior of ?-local formations. In fact, extensions and improvements of several results of Shemetkov are natural consequences of our study. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that if F is a global field, then for any integer n > 3, there is an extension field E over F of degree n such that K 2 E is not generated by the Steinberg symbols { a , b } with a ? F *, b , ? E *. If however, F is a number field and D is a finite-dimensional central division F -algebra with square free index, then K 2 D is always generated by the Steinberg symbols { a , b } with a ? F *, b ? D *. Finally, the tame kernels of central division algebras over F are expressed explicitly. [ABSTRACT FROM AUTHOR]

In the group Aut k ( A 1 ( k )) of k -automorphisms of A 1 : = A 1 (k) (the first Weyl algebra on a field k of any characteristic p = 0), we solve the following problem: Find s ? Aut k ( A 1 ), such that where 0 ? b ? k [t] and n = deg(b). This problem is a particular case of the general problem of Stafford (1987) on isomorphisms between two k -algebras &PlusMinus; b ?? and &PlusMinus; b ??' both Morita equivalent to A 1 . In this paper, we study affine algebraic curves X (b) introduced by Letzter (1992) and Perkins (1991) and their algebra of differential operators &PlusMinus; b ??( X (b)). Due to the resolution of the problem above, we find the condition to have an isomorphism between two such algebras of differential operators. In the case of isomorphism, we define an explicit isomorphism. In particular, we make explicit isomorphisms announced in Letzter (1992) and Perkins (1991). Notice that in case k is an algebraically closed field of characteristic zero, the class of algebras of differential operators &PlusMinus; b ??( X (b)) is a very important one, since any k -algebra &PlusMinus; b B Morita equivalent to A 1 is isomorphic to some &PlusMinus; b ??( X (b)) in Kouakou (2003). Dans le groupe Aut k ( A 1 (k)) des k -automorphismes de A 1 = A 1 (k) (la première algèbre de Weyl sur k , un corps commutatif de caractéristique quelconque bip ) nous résolvons le problème: Trouver s ? Aut k A 1 tel que s ( A 1 + b ) = A 1 + t n où 0 ? b ? k [t] et n = deg( b ). Ce problème est un cas particulier du problème général de Stafford (1987) sur l’existence d’isomorphisme entre deux k -algèbres &PlusMinus; b ?? et &PlusMinus; b ??' toutes les deux Morita équivalentes à A 1 . Dans ce papier, nous étudions les courbes affines X (b) introduites par Letzter (1992) et Perkins (1991) et leurs algèbres d’opérateurs diffrentiels associés &PlusMinus; b ??( X (b)). Grâce á la résolution du problème ci-dessus, nous trouvons la condition pour que deux quelconques de ces algèbres soient isomorphes. Dans le cas d’isomorphisme, nous donnons explicitement un isomorphisme. En particulier, nous explicitons les isomorphismes annoncés dans Letzter (1992) et Perkins (1991). Notons que dans le cas où k est un corps algébriquement clos de caractéristique nulle, la classe des algèbres d’opérateurs differentiels &PlusMinus; b ??( X (b)) est très importante puisque toute k -algèbre &PlusMinus; b B Morita équivalente à A 1 est isomorphe à un &PlusMinus; b ??( X (b)); see Kouakou (2003). [ABSTRACT FROM AUTHOR]

In this paper we calculate presentations for same natural monoids of transformations on a chain Xn &lcub;1 < 2 < ... < n&rcub;. First we consider OD, [PODn], the monoid of all full [partial] transformations on Xn that preserve or reverse the order. Two other monoids of partial transformations on Xn we took at are PODn and PORn—the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation. [ABSTRACT FROM AUTHOR]

In this paper we explicitly determine the derivation algebra, automorphism group of quasi Ln-filiform Lie algebras, and applying some properties of root vector decomposition we obtain an isomorphism theorem of quasi Ln-filiform Lie algebras. [ABSTRACT FROM AUTHOR]

Our work starts from the persistence index G(RlI) of the k-algebra RlI introduced by Presser (2002). It is the degree where the maximal growth of the Hilbert function begins to persist in the sense of Gotzmann. The main results of this paper consist of two parts. First, by introducing an invariant T(RlI), we show that G(RlI) is equal to the Gotzmann numbers of the Hilbert polynomial for a saturated ideal I. For a closed subscheme X in &scriptP;r and its defining ideal IX, we set G(X) as G(RlIX). Then, we present the Hilbert polynomial P(X) in terms of G(X) and inductively defined new invariants Gi(X)(0 ≤ i ≤ dim X). From this, we find a formula for C(X) = G(X) - G(X ∪ &scriptH;), where &scriptH; is a general hyperplane. In defining Gi(X), our theorem on maximal growth and the hyperplane restriction theorem are used (Theorem 3.3). In the second result, given a homogeneous saturated ideal 1, we find some conditions under which the Betti numbers of 1lex do not vanish. He focus our attention on the inequality in Green's hyperplane restriction theorem or Gotzmann numbers 'GI(X)s . Theorem 5.7 is one of the cases where the Betti numbers of 1lex do not vanish. This is a specialization of Theorem 4.5 in Presser (2002) to the situation of saturated ideals. [ABSTRACT FROM AUTHOR]

Let &scriptTx denote the full transformation semigroup on a set X. For an equivalence E on X, let TE(X) = {f ∈ &scriptTx : ∀ (a,b) ∈ E, (f(a),f(b)) ∈ E}. Then TE(X), is exactly the semigroup of continuous selfmaps of the topological space X for which the collection of all E-classes is a basis. In this paper, we discuss regularity of elements and Green's relations for TE(X). [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that char(F) ≠ 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the structure of finite groups G admitting a normal subgroup N such that N contains almost all conjugacy classes of G. [ABSTRACT FROM AUTHOR]

We generalise theory of Lawson for inverse monoids with zero to wider classes of regular semigroups. We give a structure theorem for [Lscr]-unipotent monoids with zero. Several connections between cancellative categories and 0-E-unitary semigroups are obtained as an application of the results of this paper. [ABSTRACT FROM AUTHOR]

This paper investigates coherent-like conditions and related properties that a trivial extension R ≔ A ∝ E might inherit from the ring A for some classes of modules E. It captures previous results dealing primarily with coherence, and also establishes satisfactory analogues of well-known coherence-like results on pullback constructions. Our results generate new families of examples of rings (with zerodivisors) subject to a given coherent-like condition. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the fixed subgroup of an arbitrary family of endomorphismsψi, i ∈ I, of a finitely generated free group F, is F-super-compressed. In particular, r([xcap]i∈IFixψi) ≤ r(M) for every subgroup M ≤ F containing [xcap]i∈IFixψi. This provides new evidence towards the inertia conjecture for fixed subgroups of free groups. As a corollary, we show that, in the free group of rank n, every strictly ascending chain of fixed subgroups has length at most 2n. This answers a question of Levitt. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph Δ(G) of G is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a &epsiv; cd(G). In this paper, we determine the graph Δ(G) when G is a finite simple group of exceptional Lie type. [ABSTRACT FROM AUTHOR]

In this paper, we show that σ-reducibility is preserved under joins with K. where K is the pseudovariety of semigroups in which idempotents are left zeros. Reducibility of joins with a the pseudovariety of semigroups in which idempotents are right zeros, is also considered. In this case, we were able to prove that σ- reducibility is preserved for joins with pseudovarieties verifying a certain property of cancellation. As an example involving the semidirect product. we prove that SI K is ,κ-tame, where SI stands for the pseudovariety of semilattiees. [ABSTRACT FROM AUTHOR]

Given a numerical semigroup S. let M(S) = S\{0} and (IM(S) - IM(S)) - {x &epsiv; No : x · IM(S) ⊆ IM(S)}. Define associated numerical semigroups B(S) := (M(S) - M(S)) and L(S) := Ut=i∞ (IM(S) - IM(S)). Set B0(S) - S, and for i ≥ 1, define Bi(S) :- B(Bi-1 (S). Similarly, set L0(S) - S, and for i ≤ 1, define Li(S) :- L(Li-1(S)). These constructions define two finite ascending chains of numerical semigroups S = B0(S) ⊆ B1(S) ⊆ … ⊆ BβS(S) - N0 and S - L0(S) ⊆ L1(S) ⊆ … ⊆ Lλ(S) = N0. It has been shown that not all numerical semigroups S have the property that Bi(S) ⊆ Li(S) for all i ≥ 0. In this paper, we prove that if S is a numerical semigroup with a set of generators that from a generalized arithmetic sequence, then Bi(S) ⊆ Li(S) for all i ≥ 0. Moreover, we see that this containment is not necessarily satisfied if a set of generators of S form an almost arithmetic sequence. In addition, we characterize numerical semigroups generated by generalized arithmetic sequences that satisfy other semigroup properties, such as symmetric, pseudo-symmetric, and Arf. [ABSTRACT FROM AUTHOR]

For any group G, πe (G) denotes the set of orders of its elements. If &Ohm; is a subset of positive integers, h(&Ohm;) stands for the number of isomorphism classes of finite groups G such that πe (G) = &Ohm; = πe (PGL(2,p)), where p ≥ 5 is a prime number. In this paper, we prove that h(&Ohm;) &epsiv; (1, ∞). [ABSTRACT FROM AUTHOR]

We study the problem concerning the influence of the index of maximal subgroup or the degree of primitive permutation representation of the finite simple groups on the structure of a group. Let G be a finite group and s be the index of maximal subgroup of the Monster M. In this paper, we prove that there exists an epimorphism from G to M or As if G has the primitive permutation representation of degree s, and as a consequence we prove that the Monster is determined by every s. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with characteristic different from two and S a non-empty subset of R. Suppose that θ, Φ are endomorphisms of R. An additive mapping F:R→R is called a generalized (θ, Φ)-derivation (resp. generalized Jordan (θ, Φ)-derivation) on S if there exists a (θ, Φ)-derivation d : R→R such that F(xy) = F(x)θ(y) + Φ(x)d(y) (resp. F(x²) = F(x)θ(x) + Φ(x)d(x)), holds for all x,y ε S. Suppose that U is a Lie ideal of R such that u² &epsilon U, for all u ε U. In the present paper, it is shown if θ is an automorphism of R then every generalized Jordan (θ, Φ)-derivation F on U is a generalized (θ, Φ)-derivation on U. [ABSTRACT FROM AUTHOR]