A class of regular semigroups with regular *- transversals

Let S be a regular semigroup. If there is a subsemigroup S ^* of S and a unary operation * in S satisfying: (1) x ^* ∈ S ^* \cap V_ S ^* (x) for all x∈ S; (2) (x ^* ) ^* =x for all x∈ S ^* ; (3) (x ^* y) ^* =y ^* x ^** and (xy ^* ) ^* =y ^** x ^* for all x,y∈ S, then S ^* is called a regular *- transversal of S ; if (3) is replaced with (xy) ^* =y ^* x ^* for all x,y∈ S, then S ^* is called a strongly regular *- transversal of S. In this paper we consider the class of regular semigroups with a strongly regular *- transversal. It is proved that these semigroups are P - regular semigroups. We characterize the structure of the regular semigroups with a strongly regular *- transversal.     

A generalization of the first Malcev theorem on nilpotent semigroups and nilpotency of the wreath product of semigroups

We describe all [0-]simple semigroups that are nilpotent in the sense of Malcev. This generalizes the first Malcev theorem on nilpotent (in the sense of Malcev) semigroups. It is proved that if the extended standard wreath product of semigroups is nilpotent in the sense of Malcev and the passive semigroup is not nilpotent, then the active semigroup of the wreath product is a finite nilpotent group. In addition to that, the passive semigroup is uniform periodic. Necessary and sufficient conditions are found under which the extended standard wreath product of semigroups is nilpotent in the sense of Malcev in the case where each of the semigroups of the wreath product generates a variety of finite step.     

A generalization of Ramsey's theorem for regular trees

An analog of Ramsey's theorem for regular trees is proved. The original theorem is a special case of it. It is shown that the denumerable case of Ramsey's theorem has no analog for binary trees.     

A counterexample to the generalization of Sperner's theorem

It has been conjectured that the analog of Sperner's theorem on non-comparable subsets of a set holds for arbitrary geometric lattices, namely, that the maximal number of non-comparable elements in a finite geometric lattice is max w(k), where w(k) is the number of elements of rank k. It is shown in this note that the conjecture is not true in general. A class of geometric lattices, each of which is a bond lattice of a finite graph, is constructed in which the conjecture fails to hold.     

A generalization of a theorem of Foxby

In this paper, it is proved that a commutative noetherian local ring admitting a finitely generated module of finite projective and injective dimensions with respect to a semidualizing module is Gorenstein. This result recovers a celebrated theorem of Foxby.      

A generalization of a theorem of Pringsheim

We consider noncommutative continued fractions of the form b₀ + a₁(b₁ + a₂(b₂ + a₃(…)⁻¹ c₃)⁻¹ c₂)⁻¹ c₁, (1) where a_n, b_n and c_n are elements of some Banach algebra B and b_n⁻¹ exists. Such expressions play an important role in the numerical investigation of various problems in theoretical physics and in applied mathematics, but up to now their convergence was not studied in the general case. In this paper we prove a theorem which is an extension of a wellknown theorem of Pringsheim and, in particular, guarantees the convergence of (1) under the following hypotheses:

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. As an application, we give a generalization of a theorem of van Vleck. The paper closes with an extensive bibliography.     

A generalization of a theorem of Edwards

In this note we extend a theorem of Edwards on the characterization of topological manifolds for polyhedra to a more general class of stratified spaces. We show that a cone-like space of dimension is a topological manifold if and only if the base space of every point in is a simply connected cone-like sphere.

     

A generalization of a theorem of Kruskal

LetA be a set ofk-subsets of {1,...,n} with . We determine the minimal cardinality of thel-shadow ofA. This theorem generalizes a theorem of Kruskal [10], Katona [8], Lindström [11] and other which gives a solution for the problem without the condition . Furthermore we give a necessary and sufficient condition for the uniqueness of the solution in the KKL theorem.     

A generalization of a theorem of Neumaier

In this paper, we show that for given positive integer p, the number of connected biregular graphs with exactly three distinct eigenvalues, and with smallest eigenvalue at least \(-p\), is finite.     

A generalization of a theorem of Ore

Let (K,v)$(K,v)$ be a discrete rank one valued field with valuation ring _Rv$R_v$. Let L/K$L/K$ be a finite extension such that the integral closure S   of _Rv$R_v$ in L   is a finitely generated _Rv$R_v$-module. Under a certain condition of v  -regularity, we obtain some results regarding the explicit computation of _Rv$R_v$-bases of S, thereby generalizing similar results that had been obtained for algebraic number fields in El Fadil et al. (2012) [7]. The classical Theorem of Index of Ore is also extended to arbitrary discrete valued fields. We give a simple counter example to point out an error in the main result of Montes and Nart (1992) [12] related to the Theorem of Index and give an additional necessary and sufficient condition for this result to be valid.