KERNEL IDEALS IN ALMOST DISTRIBUTIVE LATTICES

2012 ◽  
Vol 05 (02) ◽  
pp. 1250024
Author(s):  
M. Sambasiva Rao ◽  
G. C. Rao
Keyword(s):  
Distributive Lattice ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Kernel Ideal ◽  
Minimal Prime ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

The concepts of ⋆-congruences and kernel ideals are introduced in a pseudo-complemented almost distributive lattice (ADL). A necessary and sufficient condition for a congruence to become a ⋆-congruence is derived. Some equivalent conditions for every ideal of a pseudo-complemented ADL to become a kernel ideal are derived. Prime kernel ideals are characterized in terms of minimal prime ideals. Properties of kernel ideals are observed in Stone ADLs.

A class of sub-almost distributive lattices in an associate almost distributive lattice through a filter

2019 ◽  
Vol 13 (07) ◽  
pp. 2050136
Author(s):  
S. Ramesh ◽  
Jogarao Gunda
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Distributive Lattices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraic Properties ◽  
Necessary And Sufficient

In this paper, we introduce a class of sub-almost distributive lattices in an associate almost distributive lattice through a filter. We obtain several algebraic properties on the class of sub-almost distributive lattices and prove that the above class forms a distributive lattice. We derive a necessary and sufficient condition that the class to become a Boolean algebra.

σ-Ideals in a 0-1 distributive lattice

2016 ◽  
Vol 09 (01) ◽  
pp. 1650025
Author(s):  
Y. S. Pawar ◽  
S. S. Khopade
Keyword(s):  
Distributive Lattice ◽  
Distributive Lattices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complemented Lattices ◽  
Necessary And Sufficient

The notion of a [Formula: see text]-ideal in a [Formula: see text]-[Formula: see text] distributive lattice is introduced and studied. Certain classes of [Formula: see text]-[Formula: see text] distributive lattices such as normal lattices, complemented lattices, generalized Stone lattices are characterized by using the properties of [Formula: see text]-ideals. Stable ideals are defined and a necessary and sufficient condition for a [Formula: see text]-ideal to be stable is given.

scholarly journals Noetherian properties in composite generalized power series rings

2020 ◽  
Vol 18 (1) ◽  
pp. 1540-1551
Author(s):  
Jung Wook Lim ◽  
Dong Yeol Oh
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Power Series ◽  
Ordered Monoid ◽  
Power Series Rings ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

Abstract Let ({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let {{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\} . Let D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set \begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array} In this paper, we give necessary conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively ordered, and sufficient conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.

TOPOLOGICAL CHARACTERIZATION OF DUALLY NORMAL ALMOST DISTRIBUTIVE LATTICES

2012 ◽  
Vol 05 (03) ◽  
pp. 1250043
Author(s):  
G. C. Rao ◽  
N. Rafi ◽  
Ravi Kumar Bandaru
Keyword(s):  
Distributive Lattice ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Topological Characterization ◽  
Maximal Ideals ◽  
Necessary And Sufficient

A dually normal almost distributive lattice is characterized topologically in terms of its maximal ideals and prime ideals. Some necessary and sufficient conditions for the space of maximal ideals to be dually normal are obtained.

Multi-Fuzzy Complex Nilpotent Matrices

2016 ◽  
Vol 5 (4) ◽  
pp. 52-76 ◽  
Author(s):  
Asit Dey ◽  
Madhumangal Pal
Keyword(s):  
Distributive Lattice ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complex Matrix ◽  
Fuzzy Matrix ◽  
Nilpotent Matrix ◽  
Nilpotent Matrices ◽  
Necessary And Sufficient

In this paper, the concept of multi-fuzzy matrix (MFM), multi-fuzzy complex matrix (MFCM), generalized multi-fuzzy complex matrix (GMFCM), generalized multi-fuzzy complex nilpotent matrix (GMFCNM) are introduced and have shown that the set of GMFCMs form a distributive lattice. Some properties and characterizations for GMFCNMs are established and in particular, a necessary and sufficient condition for an n × n GMFCNM to have the nilpotent index n is given. Finally, the reduction of GMFCNMs over a distributive lattice are given with some properties.

scholarly journals Prime ideal characterization of chain based lattices

1980 ◽  
Vol 30 (1) ◽  
pp. 98-106
Author(s):  
U. Maddna Swamy ◽  
G. C. Rao ◽  
P. Manikyamba
Keyword(s):  
Prime Ideal ◽  
Prime Ideals ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

AbstractEpstein and Horn, in their paper ‘Chain based lattices’, characterized P1-lattices, and P2-lattices in terms of their prime ideals. But no such prime ideal characterization for P0-lattices was given. Our main aim in this paper is to characterize P0-lattices in terms of their prime ideals. We also give a necessary and sufficient condition for a P-algebra to be a P0-lattice (and hence a P2-lattice).

scholarly journals Intersections of Primary Ideals in Rings of Continuous Functions

1972 ◽  
Vol 24 (3) ◽  
pp. 502-519 ◽  
Author(s):  
R. Douglas Williams
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Prime Ideals ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Completely Regular ◽  
Rings Of Continuous Functions ◽  
Necessary And Sufficient ◽  
Primary Ideals

Let C be the ring of all real valued continuous functions on a completely regular topological space. This paper is an investigation of the ideals of C that are intersections of prime or of primary ideals.C. W. Kohls has analyzed the prime ideals of C in [3 ; 4] and the primary ideals of C in [5]. He showed that these ideals are absolutely convex. (An ideal I of C is called absolutely convex if |f| ≦ |g| and g ∈ I imply that f ∈ I.) It follows that any intersection of prime or of primary ideals is absolutely convex. We consider here the problem of finding a necessary and sufficient condition for an absolutely convex ideal I of C to be an intersection of prime ideals and the problem of finding a necessary and sufficient condition for I to be an intersection of primary ideals.

Dense elements characterization of quasi-complemented Almost Distributive Lattices

2014 ◽  
Vol 07 (04) ◽  
pp. 1450062
Author(s):  
G. C. Rao ◽  
G. Nanaji Rao ◽  
A. Lakshmana
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The properties of the set D of dense elements of an ADL are studied. The filter congruence θD generated by D in quasi-complemented ADLs is characterized. Quasi-complemented ADLs is characterized in terms of dense elements. A necessary and sufficient condition for a quasi-complemented ADLs to become a Boolean algebra is established.

scholarly journals Spaces of ideals of distributive lattices II. Minimal prime ideals

1974 ◽  
Vol 18 (1) ◽  
pp. 54-72 ◽  
Author(s):  
T. P. Speed
Keyword(s):  
Distributive Lattice ◽  
Commutative Rings ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Commutative Semigroups ◽  
Satisfactory State ◽  
Minimal Prime ◽  
The Relationship

This paper, the second of a sequence beginning with [14], deals with the relationship between a distributive lattice L = (L; ∨, ∧, 0) with zero, and certain spaces of minimal prime ideals of L. Similar studies of minimal prime ideals in commutative semigroups [8] and in commutative rings [6] inspired this work, and many of our results are similar to ones in these two articles. However the nature of our situation enables many of these results to be pushed deeper and thus to arrive at a more satisfactory state; indeed with the insight obtained from the simpler lattice situation, one can return to some topics considered in [6], [8] and give complete accounts. We do not do this in the present paper, but leave the details to the reader, see e.g. [15]. Also a study of minimal prime ideals illuminates some topics in the theory of distributive lattices, particularly Stone lattices.

UNIQUENESS OF INTEGRANDS IN QUANTUM STOCHASTIC INTEGRAL

2006 ◽  
Vol 09 (04) ◽  
pp. 607-616 ◽  
Author(s):  
UN CIG JI ◽  
KALYAN B. SINHA
Keyword(s):  
Stochastic Integral ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
General Study ◽  
Martingale Representation ◽  
Necessary And Sufficient ◽  
Quantum Stochastic Integral ◽  
Equivalent Conditions

As a general study for uniqueness of integrands in quantum martingale representation, we present a necessary and sufficient condition for uniqueness of integrands in a quantum stochastic integral. Also, several equivalent conditions to the necessary and sufficient condition are studied.

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