Two multiplicative lattice examples

1981 ◽  
Vol 12 (1) ◽  
pp. 132-133
Author(s):  
Johnny A. Johnson ◽  
Dottie Perez
Keyword(s):  
Multiplicative Lattice

scholarly journals Second and Secondary Lattice Modules

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Fethi Çallıalp ◽  
Ünsal Tekir ◽  
Emel Aslankarayiğit Uğurlu ◽  
Kürşat Hakan Oral
Keyword(s):  
Multiplicative Lattice

LetMbe a lattice module over the multiplicative latticeL. A nonzeroL-lattice moduleMis called second if for eacha∈L,a1M=1Mora1M=0M. A nonzeroL-lattice moduleMis called secondary if for eacha∈L,a1M=1Moran1M=0Mfor somen>0. Our objective is to investigative properties of second and secondary lattice modules.

scholarly journals Residuation Properties and Weakly Primary Elements in Lattice Modules

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
C. S. Manjarekar ◽  
U. N. Kandale
Keyword(s):  
Prime Element ◽  
Multiplicative Lattice ◽  
Primary Element

We obtain some elementary residuation properties in lattice modules and obtain a relation between a weakly primary element in a lattice module M and weakly prime element of a multiplicative lattice L.

scholarly journals ϕ-Prime and ϕ-Primary Elements in Multiplicative Lattices

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
C. S. Manjarekar ◽  
A. V. Bingi
Keyword(s):  
Multiplicative Lattice ◽  
Primary Element ◽  
Prime Elements ◽  
Compactly Generated ◽  
Almost Prime ◽  
Multiplicative Lattices

We investigate ϕ-prime and ϕ-primary elements in a compactly generated multiplicative lattice L. By a counterexample, it is shown that a ϕ-primary element in L need not be primary. Some characterizations of ϕ-primary and ϕ-prime elements in L are obtained. Finally, some results for almost prime and almost primary elements in L with characterizations are obtained.

Diameter and girth of the multiplicative zero-divisor graph of multiplicative lattices

2016 ◽  
Vol 09 (04) ◽  
pp. 1650071
Author(s):  
Vinayak Joshi ◽  
Sachin Sarode
Keyword(s):  
Zero Divisor ◽  
Commutative Rings ◽  
Zero Divisor Graph ◽  
Multiplicative Lattice ◽  
Minimal Prime ◽  
Prime Elements ◽  
Multiplicative Lattices

In this paper, we study the multiplicative zero-divisor graph [Formula: see text] of a multiplicative lattice [Formula: see text]. Under certain conditions, we prove that for a reduced multiplicative lattice [Formula: see text] having more than two minimal prime elements, [Formula: see text] contains a cycle and [Formula: see text]. This essentially settles the conjecture of Behboodi and Rakeei [The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10(4) (2011) 741–753]. Further, we have characterized the diameter of [Formula: see text].

scholarly journals The asymptotic and integral closure operations in multiplicative lattice modules

2005 ◽  
Vol 36 (4) ◽  
pp. 345-358
Author(s):  
Sylvia M. Foster ◽  
Johnny A. Johnson
Keyword(s):  
Chain Condition ◽  
Integral Closure ◽  
Multiplicative Lattice ◽  
Closure Operations ◽  
Ascending Chain Condition

This paper is primarily concerned with the integral and asymptotic closure operations on a multiplicative lattice relative to the greatest element of a lattice module having the ascending chain condition. We show that a cancellation law holds for the asymptotic closure of elements of the multiplicative lattice and we ultimately show, by means of multiplicative filtrations and filtration transforms, that the asymptotic closure of an element in a multiplicative lattice relative to the greatest element of a lattice module, coincides with its integral closure relative to this element in the lattice module.

scholarly journals Open Set Lattices of Subspaces of Spectrum Spaces

2015 ◽  
Vol 48 (4) ◽  
Author(s):  
Y.T. Nai ◽  
D. Zhao
Keyword(s):  
Continuous Lattice ◽  
Prime Spectrum ◽  
Unified Approach ◽  
Maximal Elements ◽  
Order Isomorphism ◽  
Open Set ◽  
Multiplicative Lattice ◽  
Radical Ideals ◽  
Prime Elements ◽  
Multiplicative Lattices

AbstractWe take a unified approach to study the open set lattices of various subspaces of the spectrum of a multiplicative lattice L. The main aim is to establish the order isomorphism between the open set lattice of the respective subspace and a sub-poset of L. The motivating result is the well known fact that the topology of the spectrum of a commutative ring R with identity is isomorphic to the lattice of all radical ideals of R. The main results are as follows: (i) for a given nonempty set S of prime elements of a multiplicative lattice L, we define the S-semiprime elements and prove that the open set lattice of the subspace S of Spec(L) is isomorphic to the lattice of all S-semiprime elements of L; (ii) if L is a continuous lattice, then the open set lattice of the prime spectrum of L is isomorphic to the lattice of all m-semiprime elements of L; (iii) we define the pure elements, a generalization of the notion of pure ideals in a multiplicative lattice and prove that for certain types of multiplicative lattices, the sub-poset of pure elements of L is isomorphic to the open set lattice of the subspace Max(L) consisting of all maximal elements of L.

scholarly journals Skeletally Dugundji spaces

2013 ◽  
Vol 11 (11) ◽  
Author(s):  
Andrzej Kucharski ◽  
Szymon Plewik ◽  
Vesko Valov
Keyword(s):  
Multiplicative Lattice ◽  
The Absolute ◽  
Skeletal Maps ◽  
Inverse Spectra

AbstractWe introduce and investigate the class of skeletally Dugundji spaces as a skeletal analogue of Dugundji space. Our main result states that the following conditions are equivalent for a given space X: (i) X is skeletally Dugundji; (ii) every compactification of X is co-absolute to a Dugundji space; (iii) every C*-embedding of the absolute p(X) in another space is strongly π-regular; (iv) X has a multiplicative lattice in the sense of Shchepin [Shchepin E.V., Topology of limit spaces with uncountable inverse spectra, Uspekhi Mat. Nauk, 1976, 31(5), 191–226 (in Russian)] consisting of skeletal maps.

M-Primary Elements of a Local Noether Lattice

1970 ◽  
Vol 22 (2) ◽  
pp. 327-331 ◽  
Author(s):  
E. W. Johnson ◽  
J. A. Johnson
Keyword(s):  
Maximal Element ◽  
Multiplicative Lattice

In this paper, we consider the extent to which a local Noether lattice (ℒ, M) is characterized by the sub-multiplicative lattice, denoted δℒ, of M-primary elements. (Here we use the notation (ℒ, M) to indicate that M is the maximal element of ℒ.) In particular, we call ℒM-complete if, given any decreasing sequence {Ai} of elements and any n ≧ 1, it follows that Ai ≦ A V Mn for large i, where A = ΛAi And we show that, given two Mi-complete local Noether lattices (ℒ1, M1) and (ℒ2, M2), with δℒ1 ≅ δℒ2, it follows that ℒ1 ≅ ℒ2. Further, we show that any local Noether lattice (ℒ, M) is a sublattice of a local Noether lattice (ℒ*, M) which is M-complete and such that δℒ = δℒ*.

Structural Properties of a New Class of CM-Lattices

1986 ◽  
Vol 38 (3) ◽  
pp. 552-562 ◽  
Author(s):  
Johnny A. Johnson ◽  
Gerald R. Sherette
Keyword(s):  
Structural Properties ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Principal Element ◽  
New Class ◽  
Multiplicative Lattice ◽  
Necessary And Sufficient ◽  
Local Results ◽  
Principal Elements ◽  
Multiplicative Lattices

1. Introduction. In this paper we introduce and study a class of multiplicative lattices called q-lattices. A q-lattice is a principally generated multiplicative lattice in which each principal element is compact. One of our main objectives is to characterize principal elements in these lattices (We note that Noether lattices and r-lattices are q-lattices [1, Theorem 2.1] and so our results apply to these two types of lattices). Among other things we determine necessary and sufficient conditions for globalizing local results in q-lattices. We then apply localization to establish some properties of principal elements in general q-lattices. Conditions equivalent to an element being principal are known for several different classes of multiplicative lattices. For example, Bogart [2] showed that if the lattice is modular, weak principal is equivalent to principal; Johnson and Lediaev pointed out that for Noether lattices, meet principal is equivalent to principal [5]; and, in an r-lattice, an element is principal if and only if it is compact and weak meet principal [6].

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