Spectral Topology on MV-Modules

2015 ◽  
Vol 11 (01) ◽  
pp. 13-33
Author(s):  
F. Forouzesh ◽  
E. Eslami ◽  
A. Borumand Saeid
Keyword(s):  
Prime Ideals ◽  
Topological Spaces ◽  
Mv Algebra ◽  
Spectral Topology

In this paper, the spectral topology and quasi-spectral topology of proper prime A-ideals in an MV-module are introduced. We show that the spectral topology of proper ⋅-prime ideals of a PMV-algebra with unity for product, is the same as the spectral topology of proper prime ideals in an MV-algebra. Also we show that the set of all prime A-ideals in an MV-module with spectral topology is not T0 and T1 topological spaces but quasi-spectral topology is T0-space and is not T1-space. Finally, we investigate when the set of all prime A-ideals in an MV-module are Hausdorff and disconnected.

scholarly journals Some results in types of extensions of MV-algebras

2019 ◽  
Vol 105 (119) ◽  
pp. 161-177
Author(s):  
F. Forouzesh ◽  
F. Sajadian ◽  
M. Bedrood
Keyword(s):  
Zero Divisor ◽  
Prime Ideals ◽  
Inverse Topology ◽  
Spectral Topology ◽  
Minimal Prime ◽  
Mv Algebras

We introduce the notions of zero divisor and extension, contraction of ideals in MV-algebras and several interesting types of extensions of MV-algebras. In particular, we show what kinds of extensions MV-algebras will lead in a homeomorphism of the spectral topology and inverse topology on minimal prime ideals. Finally, we investigate the relations among types of extensions of MV-algebras.

scholarly journals Rings which resemble rings of entire functions

1983 ◽  
Vol 24 (1) ◽  
pp. 7-16
Author(s):  
David E. Rush
Keyword(s):  
Riemann Surface ◽  
Partially Ordered Set ◽  
Riemann Surfaces ◽  
Entire Functions ◽  
Ordered Set ◽  
Ideal Theory ◽  
Prime Ideals ◽  
Topological Spaces ◽  
Valuation Theory ◽  
The Ideal

Since Helmer's 1940 paper [9] laid the foundations for the study of the ideal theory of the ring A(ℂ) of entire functions, many interesting results have been obtained for the rings A(X) of analytic functions on non-compact connected Riemann surfaces. For example, the partially ordered set Spec (A(ℂ) of prime ideals of A(ℂ) has been described by Henrikson and others [2], [10], [11]. Also, it has been shown by Ailing [4] that Spec(A(ℂ))sSpec(A(X)) as topological spaces for any non-compact connected Riemann surface X. Many results on the valuation theory of A(X) have also been obtained [1], [2]. In this note we show that a large portion of the results on the rings A(X) extend to the W-rings with complete principal divisor space which were defined by J. Klingen in [15], [16]. Therefore, many properties of A(ℂ) are shared by its non-archimedian counterparts studied by M. Lazard, M. Krasner, and others [8], [17], [18].

scholarly journals Some structure spaces of generalized semirings

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4461-4472
Author(s):  
Kostaq Hila ◽  
Sukhendu Kar ◽  
Shkëlqim Kuka ◽  
Krisanthi Naka
Keyword(s):  
Closure Operator ◽  
Prime Ideals ◽  
Topological Spaces ◽  
Inclusion Relation ◽  
Separation Axioms ◽  
Maximal Ideals ◽  
Strongly Irreducible

In this paper, we study some structure spaces of (m,n)-semiring (S,f,1), introducing the classes of n-ary prime k-ideals, n-ary prime full k-ideals, n-ary prime ideals, maximal ideals and strongly irreducible ideals. Considering their collections, respectively, of an (m,n)-semiring (S,f, 1), we construct the respective topologies on them by means of closure operator defined in terms of intersection and inclusion relation among these ideals of the (m,n)-semiring (S,f,1). The obtained topological spaces are called the structure spaces of (m,n)-semiring (S,f,1). We mainly study several principal topological axioms and properties of those structure spaces of (m,n)-semiring such as separation axioms, compactness and connectedness etc.

scholarly journals Zariski Topology

2018 ◽  
Vol 26 (4) ◽  
pp. 277-283
Author(s):  
Yasushige Watase
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Jacobson Radical ◽  
Commutative Rings ◽  
Ring Homomorphism ◽  
Ring Theory ◽  
Prime Ideals ◽  
Topological Spaces ◽  
Zariski Topology ◽  
Prime Spectrum

Summary We formalize in the Mizar system [3], [4] basic definitions of commutative ring theory such as prime spectrum, nilradical, Jacobson radical, local ring, and semi-local ring [5], [6], then formalize proofs of some related theorems along with the first chapter of [1]. The article introduces the so-called Zariski topology. The set of all prime ideals of a commutative ring A is called the prime spectrum of A denoted by Spectrum A. A new functor Spec generates Zariski topology to make Spectrum A a topological space. A different role is given to Spec as a map from a ring morphism of commutative rings to that of topological spaces by the following manner: for a ring homomorphism h : A → B, we defined (Spec h) : Spec B → Spec A by (Spec h)(𝔭) = h−1(𝔭) where 𝔭 2 Spec B.

ON ir-CLOSED SETS IN TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (5) ◽  
pp. 2573-2582
Author(s):  
A. M. Anto ◽  
G. S. Rekha ◽  
M. Mallayya
Keyword(s):  
Topological Spaces ◽  
Closed Sets

ON A NEW CLASS OF SEMI GENERALIZED CLOSED SETS IN STRONG GENERALIZED TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (11) ◽  
pp. 9353-9360
Author(s):  
G. Selvi ◽  
I. Rajasekaran
Keyword(s):  
Topological Spaces ◽  
Closed Set ◽  
Basic Properties ◽  
Closed Sets ◽  
New Class ◽  
Open Set ◽  
Continuous Maps

This paper deals with the concepts of semi generalized closed sets in strong generalized topological spaces such as $sg^{\star \star}_\mu$-closed set, $sg^{\star \star}_\mu$-open set, $g^{\star \star}_\mu$-closed set, $g^{\star \star}_\mu$-open set and studied some of its basic properties included with $sg^{\star \star}_\mu$-continuous maps, $sg^{\star \star}_\mu$-irresolute maps and $T_\frac{1}{2}$-space in strong generalized topological spaces.

Sign in / Sign up

Export Citation Format

Share Document