Page 10 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 10
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
The first chapter begins with a review of basic information about lattices and universal
algebra. This is followed by a review of results about bands. The reader already familiar with
these areas can move to the third section. (Caveat: it is imperative that one be grounded in the
basic theory of bands, and especially the theory of regular bands, to be comfortable reading this
monograph.) The third and final section provides introductory definitions and a few general
results about noncommutative lattices. The latter are always given as at algebras (S; ∧, ∨) where
∧ and ∨ are associative, idempotent binary operations that jointly satisfy a set of absorption laws.
Two varieties in particular are introduced: the variety of quasilattices and the variety of
paralattices. Their intersection, the variety of refined quasilattices, contains the variety of skew
lattices, our main topic. Placing skew lattices in a larger context provides a better sense of their
place within the pantheon of generalizations of lattices.
In Chapter 2, Skew Lattices, we proceed to study skew lattices in earnest, starting with a
selection of basic concepts and results in the first two sections. These include two decomposition
theorems (Theorems 2.1.2 and 2.1.5.) and initial results about skew lattices of idempotents in
rings. In Section 2.3 we study the important class of normal skew lattices for which
x∧y∧z∧w = x∧z∧y∧w. (If the idempotents of a ring are closed under multiplication, then they
form a normal skew lattice; indeed they form a skew Boolean algebra.) The remaining sections
provide a deeper general analysis of the structure of a skew lattice. Sections 2.1 – 2.3 of this
chapter are all that is required to read the remaining chapters, except for Sections 5.3 – 5.6.
Chapter 3 is entitled Quasilattices, Paralattices and Their Congruences. Sections 3.1 –
3.3 study congruences on quasilattices, thus obtaining results applicable to skew lattices. Section
4 looks at paralattices and especially refined quasilattices. We show the latter to be roughly skew
lattices in the sense given in Theorem 3.4.14. Section 5 discusses the effects of distributivity, the
main results being Theorems 3.5.1 and 3.5.2. Section 6 is recreational.
Skew Boolean algebras are studied in Chapter 4. Skew Boolean algebras are algebras
(S; ∧, ∨, \. 0) of type (2, 2, 2, 0) that look and behave in many ways like Boolean algebras, except
that they need not be commutative. Boolean algebras decompose at will, and so do these algebras.
This leads to a crisp description of the finitely generated (and thus finite) algebras, and of finite
free algebras in particular. The final two sections are about skew Boolean algebras for which the
natural partial order ≥ has a meet called the intersection and denoted by ∩. Many skew Boolean
algebras have intersections, e.g., all free algebras do. For partial function algebras, ∩ is the
standard set-theoretic intersection of the involved partial functions.
As noted above, the core of this monograph is these four chapters. More specialized
topics are studied in the last three chapters. Chapter 5 is entitled Further Topics in Skew Lattices,
Chapter 6 is entitled Skew Lattices in Rings and the final chapter is entitled Further Topics in
Skew Boolean Algebras.
As the reader will see, skew Boolean algebras understandably get a good bit of attention
in this monograph. There is other research on these algebras that is not in this monograph. It is
typically of more recent vintage, with much being quite good. Hopefully, before too long, some
motivated individual or group will produce a monograph devoted to skew Boolean algebras and
related topics.
8
The first chapter begins with a review of basic information about lattices and universal
algebra. This is followed by a review of results about bands. The reader already familiar with
these areas can move to the third section. (Caveat: it is imperative that one be grounded in the
basic theory of bands, and especially the theory of regular bands, to be comfortable reading this
monograph.) The third and final section provides introductory definitions and a few general
results about noncommutative lattices. The latter are always given as at algebras (S; ∧, ∨) where
∧ and ∨ are associative, idempotent binary operations that jointly satisfy a set of absorption laws.
Two varieties in particular are introduced: the variety of quasilattices and the variety of
paralattices. Their intersection, the variety of refined quasilattices, contains the variety of skew
lattices, our main topic. Placing skew lattices in a larger context provides a better sense of their
place within the pantheon of generalizations of lattices.
In Chapter 2, Skew Lattices, we proceed to study skew lattices in earnest, starting with a
selection of basic concepts and results in the first two sections. These include two decomposition
theorems (Theorems 2.1.2 and 2.1.5.) and initial results about skew lattices of idempotents in
rings. In Section 2.3 we study the important class of normal skew lattices for which
x∧y∧z∧w = x∧z∧y∧w. (If the idempotents of a ring are closed under multiplication, then they
form a normal skew lattice; indeed they form a skew Boolean algebra.) The remaining sections
provide a deeper general analysis of the structure of a skew lattice. Sections 2.1 – 2.3 of this
chapter are all that is required to read the remaining chapters, except for Sections 5.3 – 5.6.
Chapter 3 is entitled Quasilattices, Paralattices and Their Congruences. Sections 3.1 –
3.3 study congruences on quasilattices, thus obtaining results applicable to skew lattices. Section
4 looks at paralattices and especially refined quasilattices. We show the latter to be roughly skew
lattices in the sense given in Theorem 3.4.14. Section 5 discusses the effects of distributivity, the
main results being Theorems 3.5.1 and 3.5.2. Section 6 is recreational.
Skew Boolean algebras are studied in Chapter 4. Skew Boolean algebras are algebras
(S; ∧, ∨, \. 0) of type (2, 2, 2, 0) that look and behave in many ways like Boolean algebras, except
that they need not be commutative. Boolean algebras decompose at will, and so do these algebras.
This leads to a crisp description of the finitely generated (and thus finite) algebras, and of finite
free algebras in particular. The final two sections are about skew Boolean algebras for which the
natural partial order ≥ has a meet called the intersection and denoted by ∩. Many skew Boolean
algebras have intersections, e.g., all free algebras do. For partial function algebras, ∩ is the
standard set-theoretic intersection of the involved partial functions.
As noted above, the core of this monograph is these four chapters. More specialized
topics are studied in the last three chapters. Chapter 5 is entitled Further Topics in Skew Lattices,
Chapter 6 is entitled Skew Lattices in Rings and the final chapter is entitled Further Topics in
Skew Boolean Algebras.
As the reader will see, skew Boolean algebras understandably get a good bit of attention
in this monograph. There is other research on these algebras that is not in this monograph. It is
typically of more recent vintage, with much being quite good. Hopefully, before too long, some
motivated individual or group will produce a monograph devoted to skew Boolean algebras and
related topics.
8
