scholarly journals The quest for strong dualities

1995 ◽  
Vol 58 (2) ◽  
pp. 248-280 ◽  
Author(s):  
David M. Clark ◽  
Brian A. Davey
Keyword(s):  
Sufficient Conditions ◽  
Strong Duality ◽  
Necessary And Sufficient Conditions ◽  
Distributive Lattices ◽  
Necessary And Sufficient ◽  
Congruence Distributive

AbstractWe give a revised and updated exposition of the theory of full dualities initiated by Clark, Davey, Krauss and Werner, introducing the (stronger) notion of a strong duality. All known full dualities turn out to be strong. A series of theorems which provide necessary and sufficient conditions for a strong duality to exist is proved. All full dualities in the literature can be obtained from these results and many new strong dualities can be derived. In particular, we show that within congruence distributive varieties every duality can be upgraded to a strong duality. Amongst the new strong dualities are the dualities of Davey, Priestley and Werner for the varieties of pseudocomplemented distributive lattices which are either strong as they stand or can easily be made strong by the addition of partial operations to the dual structures.

Properties of Stone almost distributive lattices

2015 ◽  
Vol 08 (01) ◽  
pp. 1550011
Author(s):  
G. C. Rao ◽  
Mihret Alamneh
Keyword(s):  
Distributive Lattice ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Distributive Lattices ◽  
Basic Facts ◽  
Necessary And Sufficient

In this paper, we study different properties of Stone almost distributive lattice (Stone ADL), prove basic facts on a Stone ADL and derive a necessary and sufficient conditions for an ADL L to be a Stone ADL.

BL-ALMOST DISTRIBUTIVE LATTICES

2012 ◽  
Vol 05 (02) ◽  
pp. 1250022 ◽  
Author(s):  
G. C. Rao ◽  
Naveen Kumar Kakumanu
Keyword(s):  
Distributive Lattice ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Distributive Lattices ◽  
Necessary And Sufficient

The concept of BL-almost distributive lattice (BL-ADL) is introduced and necessary and sufficient conditions for an ADL to become a BL-ADL are derived. Different characterizations of a BL-ADL are obtained.

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.

scholarly journals Functions and equations in classes of distributive lattices with pseudocomplementation

1974 ◽  
Vol 19 (2) ◽  
pp. 191-203
Author(s):  
R. Beazer
Keyword(s):  
Special Class ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Distributive Lattices ◽  
Necessary And Sufficient

In (8), R. L. Goodstein gave necessary and sufficient conditions for the solvability of equations over distributive lattices with 0 and 1 together with an algorithm for computing a solution whenever one exists. In addition, the same problem was considered for a special class of equations over distributive lattices with pseudocomplementation. The validity of several of Goodstein's results for distributive lattices without 0 and 1 was pointed out by Rudeanu in (15) and (16).

XVIII.—The Solution of Equations in a Lattice.

1967 ◽  
Vol 67 (3) ◽  
pp. 231-242 ◽  
Author(s):  
R. L. Goodstein
Keyword(s):  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Distributive Lattices ◽  
Solution Of Equations ◽  
Solutions Of Equations ◽  
Necessary And Sufficient

SynopsisNecessary and sufficient conditions are given for the existence of solutions of equations, in any number of variables, on distributive lattices with least and greatest elements, together with an algorithm for determining a solution when these conditions are satisfied.

Characterization of BL-almost distributive lattices

2015 ◽  
Vol 08 (03) ◽  
pp. 1550041 ◽  
Author(s):  
G. C. Rao ◽  
Naveen Kumar Kakumanu
Keyword(s):  
Distributive Lattice ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Distributive Lattices ◽  
Necessary And Sufficient

Necessary and sufficient conditions for an Almost Distributive Lattice (ADL) to become a BL-ADL are derived. Different characterization of a BL-ADL are obtained. Relation between the operators on BL-ADL are derived. Wide choice of fundamental operations for a BL-ADL are derived.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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