Semidirect products and wreath products of strongly π-inverse monoids

In this paper we determine the necessary and sufficient conditions for the semidirect products and the wreath products of two monoids to be strongly -inverse. Furthermore, we determine the least group congruence on a strongly -inverse monoid, and we give some important isomorphism theorems.     

Certain free products of operator spaces

Certain free products are introduced for operator spaces and dual operator spaces. It is shown that the free product of operator spaces does not preserve the injectivity. The linking C*-algebra of the full free product of two ternary rings of operators (simply, TRO's) is *-isomorphic to the full free product of the linking C*-algebras of the two TRO's. The operator space-reduced free product of the preduals of von Neumann algebras agrees with the predual of the reduced free product of the von Neumann algebras. Each of two operator spaces can be embedded completely isometrically into the reduced free product of the operator spaces. Finally, an example is presented to show that the C*-algebra-reduced free product of two C*-algebras may be contractively isomorphic to a proper subspace of their reduced free product as operator spaces.     

Uniqueness of Rankin–Selberg products

In the present paper, we show the equality of the γ  -factors defined by Jacquet, Piatetski-Shapiro and Shalika with those obtained via the Langlands–Shahidi method. Our results are new in the case of positive characteristic, where we establish a refined version of the local–global principle for _GLn${\mathrm{GL}}_n$ which has independent interest. In characteristic zero, the results are due to Shahidi. The comparison of γ-factors is made via a uniqueness result for Rankin–Selberg γ-factors.     

Sums of products of Apostol–Bernoulli numbers

By expressing the sums of products of the Apostol?CBernoulli polynomials in terms of the special values of multiple Hurwitz?CLerch zeta functions at non-positive integers, we obtain the sums of products identity for the Apostol?CBernoulli numbers which is an analogue of the classical sums of products identity for Bernoulli numbers dating back to Euler.     

Triple products of Coleman’s families

We discuss modular forms as objects of computer algebra and as elements of certain p-adic Banach modules. We discuss a problem-solving approach in number theory, which is based on the use of generating functions and their connection with modular forms. In particular, the critical values of various L-functions of modular forms produce nontrivial but computable solutions of arithmetical problems. Namely, for a prime number we consider three classical cusp eigenforms

On the products of ℙ-subnormal subgroups of finite groups

A subgroup H of a finite group G is called ℙ-subnormal in G whenever H either coincides with G or is connected to G by a chain of subgroups of prime indices. If every Sylow subgroup of G is ℙ-subnormal in G then G is called a w-supersoluble group. We obtain some properties of ℙ-subnormal subgroups and the groups that are products of two ℙ-subnormal subgroups, in particular, of ℙ-subnormal w-supersoluble subgroups.     

Tensor products of power Köthe spaces of different types

The multirectangular characteristics µ _m ^(λ,c) are applied to the isomorphic classification of tensor products of the form $ E_0 (a)\widehat \otimes E_\infty (b) $ . We single out a subclass of tensor products such that the two-rectangular characteristic µ ₂ ^(λ,c) is a complete invariant on this class.     

Property ∏σ and tensor products of operator algebras

In this paper, we introduce Property ∏σ of operator algebras and prove that nest subalgebras and the finite-width CSL subalgebras of arbitrary von Neumann algebras have Property ∏σ.Finally, we show that the tensor product formula alg ML1-(×)algNL2 = algM-(×)N(L1 (×) L2) holds for any two finite-width CSLs L1 and L2 in arbitrary von Neumann algebras M and N, respectively.     

Double convergence and products of Fréchet spaces

The paper is devoted to convergence of double sequences and its application to products. In a convergence space we recognize three types of double convergences and points, respectively. We give examples and describe their structure and properties. We investigate the relationship between the topological and convergence closure product of two Fréchet spaces. In particular, we give a necessary and sufficient condition for the topological product of two compact Hausdorff Fréchet spaces to be a Fréchet space.     

Quasi-ideal transversals of abundant semigroups and spined products

We provide a new and much simpler structure for quasi-ideal adequate transversals of abundant semigroups in terms of spined products, which is similar in nature to that given by Saito in Proc. 8th Symposium on Semigroups, pp. 22–25 (1985) for weakly multiplicative inverse transversals of regular semigroups. As a consequence we deduce a similar result for multiplicative transversals of abundant semigroups and also consider the case when the semigroups are in fact regular and provide some new structure theorems for inverse transversals.     

Several identities for certain products of theta functions

By means of a technique used by Carlitz and Subbarao to prove the quintuple product identity (Proc. Am. Math. Soc. 32(1):42–44, 1972), we recover a general identity (Chu and Yan, Electron. J. Comb. 14:#N7, 2007) for expanding the product of two Jacobi triple products. For applications, we briefly explore identities for certain products of theta functions φ(q), ψ(q) and modular relations for the Göllnitz-Gordon functions.     

Classifying bicrossed products of two Sweedler’s Hopf algebras

We continue the study started recently by Agore, Bontea and Militaru in “Classifying bicrossed products of Hopf algebras” (2014), by describing and classifying all Hopf algebras E that factorize through two Sweedler’s Hopf algebras. Equivalently, we classify all bicrossed products H ₄ ? H ₄. There are three steps in our approach. First, we explicitly describe the set of all matched pairs (H ₄,H ₄, ?, ?) by proving that, with the exception of the trivial pair, this set is parameterized by the ground field k. Then, for any λ ∈ k, we describe by generators and relations the associated bicrossed product, \({H_{16,\lambda }}\) . This is a 16-dimensional, pointed, unimodular and non-semisimple Hopf algebra. A Hopf algebra E factorizes through H ₄ and H ₄ if and only if E ? H ₄ ? H ₄ or \(E \cong {H_{16,\lambda }}\) . In the last step we classify these quantum groups by proving that there are only three isomorphism classes represented by: H ₄ ? H ₄, H _16,0 and H _16,1 ? D(H ₄), the Drinfel’d double of H ₄. The automorphism group of these objects is also computed: in particular, we prove that Aut_Hopf (D(H ₄)) is isomorphic to a semidirect product of groups, k ^× ? ?₂.     

Algebraic properties of Zappa–Szép products of semigroups and monoids

Direct, semidirect and Zappa–Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa–Szép products of monoids and semigroups and investigate generalised Greens relations \({\mathcal R}^{*},\, {\mathcal L}^{*},\, \widetilde{\mathcal {R}}_E\) and \(\widetilde{\mathcal {L}}_E\) for these Zappa–Szép products. We consider a left restriction semigroup S with semilattice of projections E and define left and right actions of S on E and E on S, respectively, to form the Zappa–Szép product \(E \bowtie S\). We further investigate properties of \(E \bowtie S\) and show that S is a retract of \(E\bowtie S\). We also find a subset T of \(E \bowtie S\) which is left restriction.     

PropertyΠ_σand tensor products of operator algebras

In this paper,we introduce PropertyΠ_σof operator algebras and prove that nest sub- algebras and the finite-width CSL subalgebras of arbitrary von Neumann algebras have PropertyΠ_σ. Finally,we show that the tensor product formula alg_ML_1(?)alg_NL_2=alg_(M(?)N)(L_1(?)L_2) holds for any two finite-width CSLs L_1 and L_2 in arbitrary von Neumann algebras M and N,respectively.     

Chern–Simons invariants and isometric immersions of warped products

In this paper, we calculate the Chern–Simons invariants on some 3-manifolds (e.g., Berger Sphere, warped product 3-manifolds) which obtain particular features in physics. We present the condition such that Berger sphere and warped product 3-manifolds are locally conformally flat. We also give a sufficient and necessary condition such that the warped product 3-manifolds can be isometrically immersed in \mathbbR⁴{\mathbb{R}^4} . The latter condition is different from those in the earlier works of others.     

Some identities for Moore–Penrose inverses of matrix products

A group of identities are established for the Moore–Penrose inverses and the weighted Moore–Penrose inverses of matrix products AB and ABC. Some consequences and applications are also presented.