Covers and complements in the subalgebra lattice of a Boolean algebra

1989 ◽  
Vol 40 (3) ◽  
pp. 371-379
Author(s):  
Ivo Düntsch
Keyword(s):  
Boolean Algebra ◽  
General Result ◽  
Final Section ◽  
Lattice Of Subalgebras ◽  
Subalgebra Lattice

Section 1 addresses the problem of covers in Sub D, the lattice of subalgebras of a Boolean algebra; we describe those BA's in whose subalgebra lattice every element has a cover, and show that every small and separable subalgebra of P(ω) has 2ω covers in SubP(ω). Section 2 is concerned with complements and quasicomplements. As a general result it is shown that Sub D is relatively complemented if and only if D is a finite– cofinite BA. Turning to Sub P(ω), we show that no small and separable D ≤ P(ω) can be a quasicomplement. In the final section, generalisations of packed algebras are discussed, and some properties of these classes are exhibited.

The Lattice of Subalgebras of a Boolean Algebra

1962 ◽  
Vol 14 ◽  
pp. 451-460 ◽  
Author(s):  
David Sachs
Keyword(s):  
Boolean Algebra ◽  
Lattice Theory ◽  
Closure Operator ◽  
Vector Spaces ◽  
Finite Partition ◽  
Partition Lattice ◽  
Topological Vector Spaces ◽  
Boolean Space ◽  
Lattice Of Subalgebras

It is well known (1, p. 162) that the lattice of subalgebras of a finite Boolean algebra is dually isomorphic to a finite partition lattice. In this paper we study the lattice of subalgebras of an arbitrary Boolean algebra. One of our main results is that the lattice of subalgebras characterizes the Boolean algebra. In order to prove this result we introduce some notions which enable us to give a characterization and representation of the lattices of subalgebras of a Boolean algebra in terms of a closure operator on the lattice of partitions of the Boolean space associated with the Boolean algebra. Our theory then has some analogy to that of the lattice theory of topological vector spaces. Of some interest is the problem of classification of Boolean algebras in terms of the properties of their lattice of subalgebras, and we obtain some results in this direction.

scholarly journals On the lattice of subalgebras of a Boolean algebra

1972 ◽  
Vol 36 (1) ◽  
pp. 87-87
Author(s):  
G. Gr{ätzer ◽  
K. M. Koh ◽  
M. Makkai
Keyword(s):  
Boolean Algebra ◽  
Lattice Of Subalgebras

scholarly journals Some properties of the lattice of subalgebras of a Boolean algebra

1985 ◽  
Vol 32 (2) ◽  
pp. 177-193 ◽  
Author(s):  
Ivo Düntsch
Keyword(s):  
Boolean Algebra ◽  
Lattice Of Subalgebras

We investigate the structure of the lattice of subalgebras of an infinite Boolean algebra; in particular, we make a contribution to the question as to when such a lattice is simple.

On the Lattice of Subalgebras of a Boolean Algebra

1972 ◽  
Vol 36 (1) ◽  
pp. 87 ◽  
Author(s):  
G. Gratzer ◽  
K. M. Koh ◽  
M. Makkai
Keyword(s):  
Boolean Algebra ◽  
Lattice Of Subalgebras

scholarly journals On the lattice of subalgebras of a Boolean algebra

1979 ◽  
Vol 29 (4) ◽  
pp. 530-545
Author(s):  
K. P. Bhaskara Rao ◽  
M. Bhaskara Rao
Keyword(s):  
Boolean Algebra ◽  
Lattice Of Subalgebras

MARS — Its Tenth Anniversary of Operation and its Future

1980 ◽  
Vol 19 (03) ◽  
pp. 125-132
Author(s):  
G. S. Lodwick ◽  
C. R. Wickizer ◽  
E. Dickhaus
Keyword(s):  
Conceptual Development ◽  
Administrative Support ◽  
Final Section ◽  
Economic Benefits ◽  
Hospital Environment ◽  
Computer Scientist ◽  
University Of Missouri ◽  
Complete Bibliography ◽  
The University ◽  
Economic Considerations

The Missouri Automated Radiology System recently passed its tenth year of clinical operation at the University of Missouri. This article presents the views of a radiologist who has been instrumental in the conceptual development and administrative support of MARS for most of this period, an economist who evaluated MARS from 1972 to 1974 as part of her doctoral dissertation, and a computer scientist who has worked for two years in the development of a Standard MUMPS version of MARS. The first section provides a historical perspective. The second deals with economic considerations of the present MARS system, and suggests those improvements which offer the greatest economic benefits. The final section discusses the new approaches employed in the latest version of MARS, as well as areas for further application in the overall radiology and hospital environment. A complete bibliography on MARS is provided for further reading.

Introduction

2018 ◽  
Author(s):  
Peter Hopkins
Keyword(s):  
Final Section ◽  
Short Note ◽  
Current Situation ◽  
Research Projects ◽  
Everyday Lives ◽  
Early Settlement ◽  
The Everyday ◽  
Initial Research

The chapters in this collection explore the everyday lives, experiences, practices and attitudes of Muslims in Scotland. In order to set the context for these chapters, in this introduction I explore the early settlement of Muslims in Scotland and discuss some of the initial research projects that charted the settlement of Asians and Pakistanis in Scotland’s main cities. I then discuss the current situation for Muslims in Scotland through data from the 2011 Scottish Census. Following a short note about the significance of the Scottish context, in the final section, the main themes and issues that have been explored in research about Muslims in Scotland.

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