Decomposition of Goursat Matrices and Subgroups of Zm x Zn

Triangular Decomposition ◽

Given the number of subgroups of Zm x Zn, we deduce the Goursat matrix. The purpose of this paper is two-fold. A first and more concrete aim is to demonstrate that the triangular decomposition of the Goursat matrix may also be written out explicitly, and furthermore that the same is true of the inverse of these triangular factors. A second and more abstract aim provides a containment relation property between subgroups of a direct product . Namely, if U2 ≤ U1 ≤ Zm x Zn, we provide necessary and sufficient conditions for U2 ≤ U1.

Semigroup Compactifications of Direct and Semidirect Products

Canadian Mathematical Bulletin ◽

10.4153/cmb-1983-037-2 ◽

1983 ◽

Vol 26 (2) ◽

pp. 233-240 ◽

Cited By ~ 2

Author(s):

Paul Milnes

Keyword(s):

Direct Product ◽

Semidirect Product ◽

Classical Result ◽

Sufficient Conditions ◽

Direct Products ◽

Almost Periodic ◽

Semidirect Products ◽

Semigroup Compactifications ◽

AbstractA classical result of I. Glicksberg and K. de Leeuw asserts that the almost periodic compactification of a direct product S × T of abelian semigroups with identity is (canonically isomorphic to) the direct product of the almost periodic compactiflcations of S and T. Some efforts have been made to generalize this result and recently H. D. Junghenn and B. T. Lerner have proved a theorem giving necessary and sufficient conditions for an F-compactification of a semidirect product S⊗σT to be a semidirect product of compactiflcations of S and T. A different such theorem is presented here along with a number of corollaries and examples which illustrate its scope and limitations. Some behaviour that can occur for semidirect products, but not for direct products, is exposed

Compact open sets in duals and projections in L1-algebras of certain semi-direct product groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100075629 ◽

1992 ◽

Vol 111 (3) ◽

pp. 545-556 ◽

Cited By ~ 8

Author(s):

Karlheinz Gröchenig ◽

Eberhard Kaniuth ◽

Keith F. Taylor

Keyword(s):

Direct Product ◽

Sufficient Conditions ◽

Direct Products ◽

The main purpose of this paper is to study projections, that is, self-adjoint idempotents, in L1-algebras of semi-direct products G = ℝ ⋉ ℝd, d ≥ 2. We establish necessary and sufficient conditions for the existence of non-zero projections in terms of the action of ℝ on ℝd. In the cases where such projections exist, we describe minimal ones in detail.

On the Efficiency of the Direct Products of Monogenic Monoids

Algebra Colloquium ◽

10.1142/s1005386707000272 ◽

2007 ◽

Vol 14 (02) ◽

pp. 279-284 ◽

Cited By ~ 1

Author(s):

Hayrullah Ayık ◽

C. M. Campbell ◽

J. J. O'Connor

Keyword(s):

Direct Product ◽

Sufficient Conditions ◽

Direct Products ◽

We give necessary and sufficient conditions for the efficiency of the direct product of finitely many finite monogenic monoids.

Trivial action on the tensor product of finite groups

Glasgow Mathematical Journal ◽

10.1017/s0017089500007357 ◽

1988 ◽

Vol 30 (3) ◽

pp. 271-274 ◽

Cited By ~ 4

Author(s):

R. J. Higgs

Keyword(s):

Exact Sequence ◽

Tensor Product ◽

Direct Product ◽

Semidirect Product ◽

Finite Groups ◽

Sufficient Conditions ◽

Schur Multiplier ◽

Let G, H and K be finite groups such that K acts on both G and H. The action of K on G and H induces an action of K on their tensor product G ⊗ H, and we shall denote the K-stable subgroup of G ⊗ H by (G ⊗ H)K. In section 1 of this note we shall obtain necessary and sufficient conditions for (G ⊗ H)K = G ⊗ H. The importance of this result is that the direct product of G and H has Schur multiplier M(G × H) isomorphic to M(G) × M(H) × (G ⊗ H); moreover K: acts on M(G × H), and M(G × H)K is one of the terms contained in a fundamental exact sequence concerning the Schur multiplier of the semidirect product of K and G × H (see [3, (2.2.10) and (2.2.5)] for details). Indeed in section 2 we shall assume that G is abelian and use the fact that M(G) ≅ G ∧ G to find necessary and sufficient conditions for M(G)K = M(G).

The Direct Decomposition of l-algebras into Products of Subdirectly Irreducible Factors

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700003463 ◽

2003 ◽

Vol 75 (1) ◽

pp. 41-56 ◽

Cited By ~ 2

Author(s):

Sándor Radeleczki

Keyword(s):

Direct Product ◽

Present Author ◽

Sufficient Conditions ◽

Direct Decomposition ◽

Subdirectly Irreducible ◽

Bounded Lattice ◽

Structure Theorems ◽

AbstractGeneralizing earlier results of Katriňák, El-Assar and the present author we prove new structure theorems for l-algebras. We obtain necessary and sufficient conditions for the decomposition of an arbitrary bounded lattice into a direct product of (finitely) subdirectly irreducible lattices.

Ideals of direct products of rings

Asian-European Journal of Mathematics ◽

10.1142/s1793557118500948 ◽

2018 ◽

Vol 11 (04) ◽

pp. 1850094

Author(s):

Ivan Chajda ◽

Günther Eigenthaler ◽

Helmut Länger

Keyword(s):

Direct Product ◽

Sufficient Conditions ◽

Commutative Rings ◽

Direct Products ◽

Type Condition ◽

It is known that an ideal of a direct product of commutative unitary rings is directly decomposable into ideals of the corresponding factors. We show that this does not hold in general for commutative rings and we find necessary and sufficient conditions for direct decomposability of ideals. For varieties of commutative rings, we derive a Mal’cev type condition characterizing direct decomposability of ideals and we determine explicitly all varieties satisfying this condition.

On separability properties in direct products of semigroups

Monatshefte für Mathematik ◽

10.1007/s00605-021-01570-4 ◽

2021 ◽

Author(s):

Gerard O’Reilly ◽

Martyn Quick ◽

Nik Ruškuc

Keyword(s):

Direct Product ◽

Sufficient Conditions ◽

Direct Products ◽

Residual Finiteness ◽

Finiteness Conditions ◽

Finite Semigroups ◽

AbstractWe investigate four finiteness conditions related to residual finiteness: complete separability, strong subsemigroup separability, weak subsemigroup separability and monogenic subsemigroup separability. For each of these properties we examine under which conditions the property is preserved under direct products. We also consider if any of the properties are inherited by the factors in a direct product. We give necessary and sufficient conditions for finite semigroups to preserve the properties of strong subsemigroup separability and monogenic subsemigroup separability in a direct product.

THE p-COCKCROFT PROPERTY OF THE SEMI-DIRECT PRODUCTS OF MONOIDS

International Journal of Algebra and Computation ◽

10.1142/s0218196703001298 ◽

2003 ◽

Vol 13 (01) ◽

pp. 1-16 ◽

Cited By ~ 11

Author(s):

A. SINAN ÇEVIK

Keyword(s):

Direct Product ◽

Sufficient Conditions ◽

Direct Products ◽

Necessary And Sufficient ◽

Standard Presentation

The semi-direct product of arbitrary two monoids and a presentation for this product have received considerable attention, see for instance [12, 14, 15]. In [15], Wang defined a trivializer set of the Squier complex associated with this presentation. In this paper, as a main result, we discuss necessary and sufficient conditions for the standard presentation of the semi-direct product of any two monoids to be p-Cockcroft for any prime p or 0. Finally we present some applications of this main theorem.

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

Direct products of automatic semigroups

10.1017/s1446788700001816 ◽

2000 ◽

Vol 69 (1) ◽

pp. 19-24 ◽

Cited By ~ 9

Author(s):

C. M. Campbell ◽

E. F. Robertson ◽

N. Ruškuc ◽

R. M. Thomas

Keyword(s):

Direct Product ◽

Sufficient Conditions ◽

Direct Products ◽

Finitely Generated ◽

Automatic Groups ◽

AbstractIt is known that the direct product of two automatic groups is automatic. The notion of automaticity bas been extended to semigroups, and this for groups has been generalized to automatic monoids. However, the direct product of two automatic semigroups need not be finitely generated and hence not automatic.Robertson, Ruškuc and Wiegold have determined necessary and sufficient conditions for the direct product of two finitely generated semigroups to be finitely generated. Building on this, we prove the following. Let S and T be automatic semigroups; if S and T are infinite, then S × T is automatic if and only if S2 = S and T2 = T; if S is finite and T is infinite, then S × T is automatic if and only if S2 = S. As a consequence, we have that, if S and T are automatic semigroups, then S × T is automatic if and only if S × T is finitely generated.

The extinction time of a general birth and death process with catastrophes

Journal of Applied Probability ◽

10.1017/s0021900200116031 ◽

1986 ◽

Vol 23 (04) ◽

pp. 851-858 ◽

Cited By ~ 1

Author(s):

P. J. Brockwell

Keyword(s):

Laplace Transform ◽

Size Distribution ◽

Sufficient Conditions ◽

Death Process ◽

Extinction Time ◽

Birth And Death Process ◽

Different Types ◽

The Laplace Transform ◽

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.