The lattice of congruences on direct products of cyclic semigroups and certain other semigroups

1983 ◽  
Vol 95 (3-4) ◽  
pp. 203-214 ◽  
Author(s):  
D. C. Trueman
Keyword(s):  
Direct Product ◽  
Ideal Extension ◽  
Semimodular Lattice ◽  
Direct Products ◽  
Upper Semimodular Lattice ◽  
Lattice Of Congruences ◽  
Finite Direct Product

SynopsisLet W be a semigroup with W\W2 non-empty, such that if ρ is a congruence on W with xpy for all x, y= W\W2, then zpw for all z, w= W2. We prove that the lattice of congruences on W is directly indecomposable, and conclude that a direct product of cyclic semigroups, with at least two non-group direct factors, has a directly indecomposable lattice of congruences. We find that the lattice of congruences on a direct product S1×S2×V of two non-trivial cyclic semigroups S1 and S2, one not being a group, and any other semigroup V, is not lower semimodular, and hence, not modular. We then prove that any finite ideal extension of a group by a nil semigroup has an upper semimodular lattice of congruences, and conclude that a finite direct product of finite cyclic semigroups has an upper semimodular lattice of congruences.

Lattices of congruences on free finitely generated commutative semigroups and direct products of cyclic monoids

1985 ◽  
Vol 100 (1-2) ◽  
pp. 175-179 ◽  
Author(s):  
D. C. Trueman
Keyword(s):  
Direct Product ◽  
Commutative Semigroup ◽  
Direct Products ◽  
Finitely Generated ◽  
Commutative Semigroups ◽  
Lattice Of Congruences

SynopsisLet S11× V1 be a direct product of a cyclic monoid S11with S1 not a group, and a semigroup V1 with an adjoined identity. We prove thatboth the lattice of congruences on (S11 × V1)/{(1,1)} and the lattice of congruences on (S11 × V1) are neither lower semimodular nor uppersemimodular. We then prove that the lattice of congruences on a free finitely generated commutative semigroup with more than one generator is neither lower semimodular nor upper semimodular.

scholarly journals ON NEAR-RING IDEMPOTENTS AND POLYNOMIALS ON DIRECT PRODUCTS OF Ω-GROUPS

2001 ◽  
Vol 44 (2) ◽  
pp. 379-388 ◽  
Author(s):  
Erhard Aichinger
Keyword(s):  
Direct Product ◽  
Subject Classification ◽  
Idempotent Element ◽  
Direct Products ◽  
Primary 16Y30 ◽  
Secondary 08A40

AbstractLet $N$ be a zero-symmetric near-ring with identity, and let $\sGa$ be a faithful tame $N$-group. We characterize those ideals of $\sGa$ that are the range of some idempotent element of $N$. Using these idempotents, we show that the polynomials on the direct product of the finite $\sOm$-groups $V_1,V_2,\dots,V_n$ can be studied componentwise if and only if $\prod_{i=1}^nV_i$ has no skew congruences.AMS 2000 Mathematics subject classification: Primary 16Y30. Secondary 08A40

scholarly journals Vertex-Transitive Direct Products of Graphs

10.37236/6999 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Richard H. Hammack ◽  
Wilfried Imrich
Keyword(s):  
Direct Product ◽  
Direct Products ◽  
Odd Cycle ◽  
Vertex Transitive ◽  
Odd Cycles

It is known that for graphs $A$ and $B$ with odd cycles, the direct product $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. But this is not necessarily true if one of $A$ or $B$ is bipartite, and until now there has been no characterization of such vertex-transitive direct products. We prove that if $A$ and $B$ are both bipartite, or both non-bipartite, then $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. Also, if $A$ has an odd cycle and $B$ is bipartite, then $A\times B$ is vertex-transitive if and only if both $A\times K_2$ and $B$ are vertex-transitive.

Topologies of Lattice Products

1966 ◽  
Vol 18 ◽  
pp. 1004-1014 ◽  
Author(s):  
Richard A. Alo ◽  
Orrin Frink
Keyword(s):  
Direct Product ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Order Relation ◽  
Direct Products ◽  
Ordered Sets ◽  
Partially Ordered

A number of different ways of defining topologies in a lattice or partially ordered set in terms of the order relation are known. Three of these methods have proved to be useful and convenient for lattices of special types, namely the ideal topology, the interval topology, and the new interval topology of Garrett Birkhoff. In another paper (2) we have shown that these three topologies are equivalent for chains (totally ordered sets), where they reduce to the usual intrinsic topology of the chain.Since many important lattices are either direct products of chains or sublattices of such products, it is natural to ask what relationships exist between the various order topologies of a direct product of lattices and those of the lattices themselves.

Semigroup Compactifications of Direct and Semidirect Products

1983 ◽  
Vol 26 (2) ◽  
pp. 233-240 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Direct Product ◽  
Semidirect Product ◽  
Classical Result ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Products ◽  
Almost Periodic ◽  
Semidirect Products ◽  
Semigroup Compactifications ◽  
Necessary And Sufficient

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

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

Direct product decomposition of theories of modules

1979 ◽  
Vol 44 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Steven Garavaglia
Keyword(s):  
Direct Product ◽  
Regular Ring ◽  
Decomposition Theorem ◽  
Direct Products ◽  
Product Decomposition ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Direct Product Decomposition ◽  
Von Neumann Regular ◽  
Complete Theories

This paper is mainly concerned with describing complete theories of modules by decomposing them (up to elementary equivalence) into direct products of simpler modules. In §1, I give a decomposition theorem which works for arbitrary direct product theories T. Given such a T, I define T-indecomposable structures and show that every model of T is elementarily equivalent to a direct product of T-indecomposable models of T. In §2, I show that if R is a commutative ring then every R-module is elementarily equivalent to ΠMM where M ranges over the maximal ideals of R and M is the localization of at M. This is applied to prove that if R is a commutative von Neumann regular ring and TR is the theory of R-modules then the TR-indecomposables are precisely the cyclic modules of the form R/M where M is a maximal ideal. In §3, I use the decomposition established in §2 to characterize the ω1-categorical and ω-stable modules over a countable commutative von Neumann regular ring and the superstable modules over a commutative von Neumann regular ring of arbitrary cardinality. In the process, I also prove several general characterizations of ω-stable and superstable modules; e.g., if R is any countable ring, then an R-moduIe is ω-stable if and only if every R-module elementarily equivalent to it is equationally compact.

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

1992 ◽  
Vol 111 (3) ◽  
pp. 545-556 ◽  
Author(s):  
Karlheinz Gröchenig ◽  
Eberhard Kaniuth ◽  
Keith F. Taylor
Keyword(s):  
Direct Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Products ◽  
Necessary And Sufficient

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

2007 ◽  
Vol 14 (02) ◽  
pp. 279-284 ◽  
Author(s):  
Hayrullah Ayık ◽  
C. M. Campbell ◽  
J. J. O'Connor
Keyword(s):  
Direct Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Products ◽  
Necessary And Sufficient

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

scholarly journals Sectional algebras of semigroupoid bundles

2020 ◽  
Vol 30 (06) ◽  
pp. 1257-1304
Author(s):  
Luiz Gustavo Cordeiro
Keyword(s):  
Direct Product ◽  
Tensor Products ◽  
Crossed Product ◽  
Direct Products ◽  
Crossed Products ◽  
Graded Algebras ◽  
Semidirect Products ◽  
Skew Products ◽  
Definition Of ◽  
Product Bundles

In this paper, we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ([Formula: see text]-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions — via the construction of a sectional algebra — are widely generalized: Direct products bundles by semigroupoids correspond to tensor products of algebras; semidirect products of bundles correspond to “naïve” crossed products of algebras; skew products of graded bundles correspond to smash products of graded algebras; Quotient bundles correspond to quotient algebras. Moreover, most of the results hold in the non-Hausdorff setting. In the course of this work, we generalize the definition of smash products to groupoid graded algebras. As an application, we prove that whenever [Formula: see text] is a ∧-preaction of a discrete inverse semigroupoid [Formula: see text] on an ample (possibly non-Hausdorff) groupoid [Formula: see text], the Steinberg algebra of the associated groupoid of germs is naturally isomorphic to a crossed product of the Steinberg algebra of [Formula: see text] by [Formula: see text]. This is a far-reaching generalization of analogous results which had been proven in particular cases.

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