Multiplicative lattices in which every principal element is a product of prime elements

1978 ◽  
Vol 8 (1) ◽  
pp. 330-335 ◽  
Author(s):  
D. D. Anderson
Keyword(s):  
Principal Element ◽  
Prime Elements ◽  
Multiplicative Lattices

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].

s-Prime elements in multiplicative lattices

1995 ◽  
Vol 31 (3) ◽  
pp. 201-208 ◽  
Author(s):  
C. Jayaram ◽  
E. W. Johnson
Keyword(s):  
Prime Elements ◽  
Multiplicative Lattices

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.

Zariski topology on lattice modules

2015 ◽  
Vol 08 (04) ◽  
pp. 1550066 ◽  
Author(s):  
Sachin Ballal ◽  
Vilas Kharat
Keyword(s):  
New York ◽  
Spectral Theory ◽  
Zariski Topology ◽  
Topological Properties ◽  
Algebraic Properties ◽  
Prime Elements ◽  
Multiplicative Lattices

Let [Formula: see text] be a lattice module over a [Formula: see text]-lattice [Formula: see text] and [Formula: see text] be the set of all prime elements in lattice modules [Formula: see text]. In this paper, we study the generalization of the Zariski topology of multiplicative lattices [N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II: Minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math. 52 (1988) 53–67; N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, in Algebra and Its Applications (Marcel Dekker, New York, 1984), pp. 265–276.] to lattice modules. Also we investigate the interplay between the topological properties of [Formula: see text] and algebraic properties of [Formula: see text].

Weakly Prime Elements in Multiplicative Lattices

2012 ◽  
Vol 40 (8) ◽  
pp. 2825-2840 ◽  
Author(s):  
Fethi Çallıalp ◽  
C. Jayaram ◽  
Ünsal Tekir
Keyword(s):  
Prime Elements ◽  
Multiplicative Lattices

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].

Materialism and Legal Historiography, from Bachelard to Benjamin

2019 ◽  
pp. 1-19
Author(s):  
Christopher Tomlins
Keyword(s):  
Walter Benjamin ◽  
Historical Materialism ◽  
First Century ◽  
Gaston Bachelard ◽  
Spatial Justice ◽  
Legal Studies ◽  
New Materialisms ◽  
The World ◽  
Prime Elements ◽  
Cultural Turn

As the linguistic/cultural turn of the last fifty years has begun to ebb, sociolegal and legal-humanist scholarship has seen an accelerating return to materiality. This chapter asks what relationship may be forthcoming between the “new materialisms” and “vibrant matter” of recent years, and the older materialisms—both historical and literary, both Marxist and non-Marxist—that held sway prior to post-structuralism. What impact might such a relationship have on the forms, notably “spatial justice,” that materiality is assuming in contemporary legal studies? To attempt answers, the chapter turns to two figures from more than half a century ago: Gaston Bachelard—once famous, now mostly forgotten; and Walter Benjamin—once largely forgotten, now famous. A prolific and much-admired writer between 1930 and 1960, Bachelard pursued two trajectories of inquiry: a dialectical and materialist and historical (but non-Marxist) philosophy of science; and a poetics of the material imagination based on inquiry into the literary reception and representation of the prime elements—earth, water, fire, and air. Between the late 1920s and 1940, meanwhile, Benjamin developed an idiosyncratic but potent form of historical materialism dedicated to “arousing [the world] from its dream of itself.” The chapter argues that by mobilizing Bachelard and Benjamin for scholarship at the intersection of law and the humanities, old and new materialisms can be brought into a satisfying conjunction that simultaneously offers a poetics for spatial justice and lays a foundation for a materialist legal historiography for the twenty-first century.

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