Page 9 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 9
Foreword
The last 30 years have witnessed sustained research by a number of individuals in skew
lattices, a class of noncommutative generalizations of lattices. (A partial list of published work is
given in the Bibliography at the end of this monograph.) Papers on noncommutative lattices in
general have appeared since the late 1940s. It would not be unfair to say that the more recent
research has been both deeper and more fruitful than the earlier work, for several reasons.
To begin, by restricting attention to a particular class of algebras, one is more focused.
Indeed, once commutativity is dropped, the possibilities for differing absorption identities that
reduce to the familiar identities in the commutative case becomes quite large. Thus, by working
within the boundaries of a fixed set of identities, one becomes more concentrated in one’s efforts.
Secondly, in more recent times advantage has been made of results in semigroup theory
about bands, that is, semigroups consisting entirely of idempotents. Indeed the newer research
began by studying multiplicative bands of idempotents in rings, and realizing that under certain
conditions such bands would also be closed under an “upward multiplication” to yield a skew
lattice. Parallel to this was an expanding role of universal algebra, both due to results of a fairly
general scope (basic universal algebra) and also results related to structures that were weakened
or modified forms of Boolean algebras. This was especially important in the study of skew
Boolean algebras. Summing up: there has been a greater awareness of relevant information.
Thirdly, as indicated above, the newer research began with a rich source of motivating
examples – bands of idempotents in rings. In particular for bands that were left regular (xyx = xy),
any maximal such band in a ring was also closed under the circle operation x¡y = x + y – xy. And
any band closed under both operations satisfied certain absorption identities, e.g.,
e(e¡f) = e = e¡(ef). These observations, along with others related to normal bands of idempotents
(that were middle commutative: xyzw = xzyw) indicated the presence of structurally enhanced
bands with a roughly lattice-like structure. Thus skew lattices arose, along with a number of
potential properties first observed in the setting of rings. To this was added a second class of
motivating examples, algebras of partial functions P(A, B) between pairs of sets, A and B. These
provided examples of skew Boolean algebras and related structures, much as “partial sets” (that is,
subsets) led to basic examples of Boolean algebras and distributive lattices.
In addition, there was the effect of computer technology, especially beginning in the late
90s. The internet provided a quick, efficient means of communication, making it easier for like-
minded individuals to connect. And computer software made it easier to find examples and initial
proofs for some theorems. This continued to impact the development of skew lattice theory in the
21st century.
And finally and most fortunately, skew lattices have attracted the attention of a number
very fine mathematicians from around the world, including (to my current awareness) individuals
from America, Australia, China, Ethiopia, Europe (including Great Britain J), India and Iran.
This present volume is an organized presentation of much that has been published on the
subject up through 2017. It is divided into seven chapters, the first four of which form the core of
this survey.
7
The last 30 years have witnessed sustained research by a number of individuals in skew
lattices, a class of noncommutative generalizations of lattices. (A partial list of published work is
given in the Bibliography at the end of this monograph.) Papers on noncommutative lattices in
general have appeared since the late 1940s. It would not be unfair to say that the more recent
research has been both deeper and more fruitful than the earlier work, for several reasons.
To begin, by restricting attention to a particular class of algebras, one is more focused.
Indeed, once commutativity is dropped, the possibilities for differing absorption identities that
reduce to the familiar identities in the commutative case becomes quite large. Thus, by working
within the boundaries of a fixed set of identities, one becomes more concentrated in one’s efforts.
Secondly, in more recent times advantage has been made of results in semigroup theory
about bands, that is, semigroups consisting entirely of idempotents. Indeed the newer research
began by studying multiplicative bands of idempotents in rings, and realizing that under certain
conditions such bands would also be closed under an “upward multiplication” to yield a skew
lattice. Parallel to this was an expanding role of universal algebra, both due to results of a fairly
general scope (basic universal algebra) and also results related to structures that were weakened
or modified forms of Boolean algebras. This was especially important in the study of skew
Boolean algebras. Summing up: there has been a greater awareness of relevant information.
Thirdly, as indicated above, the newer research began with a rich source of motivating
examples – bands of idempotents in rings. In particular for bands that were left regular (xyx = xy),
any maximal such band in a ring was also closed under the circle operation x¡y = x + y – xy. And
any band closed under both operations satisfied certain absorption identities, e.g.,
e(e¡f) = e = e¡(ef). These observations, along with others related to normal bands of idempotents
(that were middle commutative: xyzw = xzyw) indicated the presence of structurally enhanced
bands with a roughly lattice-like structure. Thus skew lattices arose, along with a number of
potential properties first observed in the setting of rings. To this was added a second class of
motivating examples, algebras of partial functions P(A, B) between pairs of sets, A and B. These
provided examples of skew Boolean algebras and related structures, much as “partial sets” (that is,
subsets) led to basic examples of Boolean algebras and distributive lattices.
In addition, there was the effect of computer technology, especially beginning in the late
90s. The internet provided a quick, efficient means of communication, making it easier for like-
minded individuals to connect. And computer software made it easier to find examples and initial
proofs for some theorems. This continued to impact the development of skew lattice theory in the
21st century.
And finally and most fortunately, skew lattices have attracted the attention of a number
very fine mathematicians from around the world, including (to my current awareness) individuals
from America, Australia, China, Ethiopia, Europe (including Great Britain J), India and Iran.
This present volume is an organized presentation of much that has been published on the
subject up through 2017. It is divided into seven chapters, the first four of which form the core of
this survey.
7
