Skew lattice
Encyclopedia
In abstract algebra
, a skew lattice is an algebraic structure
that is a non-commutative generalization of a lattice
. While the term skew lattice can be used to refer to any non-commutative generalization of a lattice, over the past twenty years it has been used primarily as follows.
s
and 
, called meet and join, that satisfy the following dual pair of absorption laws
and 
.
Given that
and 
are associative and idempotent, these identities are equivalent to the dualities:
iff 
and 
iff 
.
and Boolean algebra; and for others it has been the behavior of idempotents in rings
. A noncommutative lattice, generally speaking, is an algebra
where 
and 
are associative, idempotent binary
operations
connected by absorption identities guaranteeing that
in some way dualizes 
. The precise identities chosen depends upon the underlying motivation, with differing choices producing distinct varieties of algebras
. Pascual Jordan, motivated by questions in quantum logic
, initiated a study of noncommutative lattices in his 1949 paper, Über Nichtkommutative Verbande, choosing the absorption identities
He referred to those algebras satisfying them as Schrägverbände. By varying or augmenting these identities, Jordan and others obtained a number of varieties of noncommutative lattices.
Beginning with Jonathan Leech's 1989 paper, Skew lattices in rings, skew lattices as defined above have been the primary objects of study. This was aided by previous results about bands. This was especially the case for many of the basic properties.
In a skew lattice
, the natural partial order is defined by 
if 
, or dually, 
. The natural preorder
on
is given by 
if 
or dually 
. While 
and 
agree on lattices, 
properly refines 
in the noncommutative case. The induced natural equivalence
is defined by 
if 
, that is, 
and
or dually, 
and 
. The blocks of the partition 
are
lattice ordered by
iff 
and 
exist such that 
. This permits us to write Hasse diagram
s of skew lattices such as the following pair:
E.g., in the diagram on the left above, that
and 
are 
related is expressed by the dashed
segment. The slanted lines reveal the natural partial order between elements of the distinct
-classes. The elements 
, 
and 
form the singleton 
-classes.
Rectangular Skew Lattices
Skew lattices consisting of a single
-class are called rectangular. They are characterized by the equivalent identities: 
, 
and 
. Rectangular skew lattices are isomorphic to skew lattices having the following construction (and conversely): given nonempty
sets
and 
, on 
define 
and 
. The 
-class partition of a skew lattice 
, as indicated in the above diagrams, is the unique partition of 
into its maximal rectangular subalgebras, Moreover, 
is a congruence with the induced quotient algebra 
being the maximal lattice image of 
, thus making every skew lattice 
a lattice of rectangular subalgebras. This is the Clifford-McLean Theorem for skew lattices, first given for bands separately by Clifford and McLean. It is also known as the First Decomposition Theorem for skew lattices.
Right (left) handed skew lattices and the Kimura factorization
A skew lattice is right-handed if it satisfies the identity
or dually, 
.
These identities essentially assert that
and 
in each 
-class. Every skew lattice 
has a unique maximal right-handed image 
where the congruence 
is defined by 
if both 
and 
(or dually, 
and 
). Likewise a skew lattice is left-handed if 
and 
in each 
-class. Again the maximal left-handed image of a skew lattice 
is the image 
where the congruence 
is defined in dual fashion to 
. Many examples of skew lattices are either right or left-handed. In the lattice of congruences, 
and 
is the identity congruence 
. The induced epimorphism 
factors through both induced epimorphisms 
and 
. Setting 
, the homomorphism 
defined by 
, induces an isomorphism 
. This is the Kimura factorization of 
into a fibred product of its maximal right and left-handed images.
Like the Clifford-McLean Theorem, Kimura factorization (or the Second Decomposition Theorem for skew lattices) was first given for regular bands (that satisfy the middle absorption
identity,
). Indeed both 
and 
are regular band operations. The above symbols 
, 
and 
come, of course, from basic semigroup theory.
For more details on the basic properties of a skew lattice please read and .
more.
Symmetric Skew Lattices
A skew lattice S is symmetric if for any
, 
iff 
. Occurrences of commutation are thus unambiguous for such skew lattices, with subsets of pairwise commuting elements generating commutative subalgebras, i.e, sublattices. ( This is not true for skew lattices in general.) Equational bases for this subvariety, first given by Spinks are:
and 
.
A lattice section of a skew lattice
is a sublattice 
of 
meeting each 
-class of 
at a single element. 
is thus an internal copy of the lattice 
with the composition 
being an isomorphism. All symmetric skew lattices for which |S/D| \leq \aleph_0 , admit a lattice section. Symmetric or not, having a lattice section 
guarantees that 
also has internal copies of 
and 
given respectively by 
and 
, where 
and 
are the 
and 
congruence classes of 
in 
. Thus 
and 
are isomorphisms (See ). This leads to a commuting diagram of embedding dualizing the preceding Kimura diagram.
Cancellative Skew Lattices
A skew lattice is cancellative if
and 
implies 
and likewise 
and 
implies 
. Cancellatice skew lattices are symmetric and can be shown to form a variety. Unlike lattices, they need not be distributive, and conversely.
Distributive Skew Lattices
Distributive skew lattices are determined by the identities:
(D1 )
(D’1 )
Unlike lattices, (D1 ) and (D‘1 ) are not equivalent in general for skew lattices, but they are for symmetric skew lattices. (See,,.) The condition (D1 ) can be strengthened to
(D2 ) in which case (D‘1 ) is a consequence. A skew lattice 
satisfies both (D2) and its dual, 
, if and only if it factors as the product of a distributive lattice and a rectangular skew lattice. In this latter case (D2 ) can be strengthened to 
and 
. (D3 )
On its own, (D3 ) is equivalent to (D2 ) when symmetry is added. (See.) We thus have six subvarieties of skew lattices determined respectively by (D1), (D2), (D3) and their duals.
Normal Skew Lattices
As seen above,
and 
satisfy the identity 
. Bands satisfying the stronger identity, 
, are called normal. A skew lattice is normal skew if it satisfies
For each element a in a normal skew lattice
, the set 
defined by { 
} or equivalently {
} is a sublattice of 
, and conversely. (Thus normal skew lattices have also been called local lattices.) When both 
and 
are normal, 
splits isomorphically into a product 
of a lattice 
and a rectangular skew lattice 
, and conversely. Thus both normal skew lattices and split skew lattices form varieties. Returning to distribution, 
so that 
characterizes the variety of distributive, normal skew lattices, and (D3) characterizes the variety of symmetric, distributive, normal skew lattices.
Categorical Skew Lattices
A skew lattice is categorical if nonempty composites of coset bijections are coset bijections. Categorical skew lattices form a variety. Skew lattices in rings and normal skew lattices are examples
of algebras on this variety. Let
with 
, 
and 
, 
be the coset bijection from 
to 
taking 
to 
, 
be the coset bijection from 
to 
taking 
to 
and finally 
be the coset bijection from 
to 
taking 
to 
. A skew lattice 
is categorical if one always has the equality 
, ie. , if the
composite partial bijection
if nonempty is a coset bijection from a 
-coset of 
to an 
-coset
of
. That is 
.
All distributive skew lattices are categorical. Though symmetric skew lattices might not be. In a sense they reveal the independence between the properties of symmetry and distributivity.
For more details on these and other subvarieties of skew lattices please read and .
, 
or, dually, 
. (0)
A Boolean skew lattice is a symmetric, distributive normal skew lattice with 0,
, such that 
is a Boolean lattice for each 
. Given such skew lattice S , a difference operator \ is defined on by x\ y = 
where the latter is evaluated in the Boolean lattice 
. In the presence of (D3) and (0), \ is characterized by the identities:
and 
One thus has a variety of skew Boolean algebras
characterized by identities (D3), (0) and (S B). A primitive skew Boolean algebra consists of 0 and a single non-0 D-class. Thus it is the result of adjoining a 0 to a rectangular skew lattice D via (0) with 
, if 
and
otherwise. Every skew Boolean algebra is a subdirect product of primitive algebras. Skew Boolean algebras play an important role in the study of discriminator varieties and other generalizations in universal algebra of Boolean behavior. For more details on skew Boolean algebras see,,,,, ,,, and.
be a Ring
and let
denote the set of all Idempotents in 
. For all 
set 
and 
.
Clearly
but also 
is associative. If a subset 
is closed under 
and 
, then 
is a distributive, cancellative skew lattice. To find such skew lattices in 
one looks at bands in 
, especially the ones that are maximal with respect to some constraint. In fact, every multiplicative band in 
that is maximal with respect to being right regular (= ) is also closed under 
and so forms a right-handed skew lattice. In general, every right regular band in 
generates a right-handed skew lattice in 
. Dual remarks also hold for left regular bands (bands satisfying the identity 
) in 
. Maximal regular bands need not to be closed under 
as defined; counterexamples are easily found using multiplicative rectangular bands. These cases are closed, however, under the cubic variant of 
defined by 
since in these cases 
reduces to 
to give the dual rectangular band. By replacing the condition of regularity by normality 
, every maximal normal multiplicative band 
in 
is also closed under 
with 
, where 
, forms a Boolean skew lattice. When 
itself is closed under multiplication, then it is a normal band and thus forms a Boolean skew lattice. In fact, any skew Boolean algebra can be embedded into such an algebra. (See.) When A has a multiplicative identity 
, the condition that 
is multiplicatively closed is well-known to imply that 
forms a Boolean algebra. Skew lattices in rings continue to be a good source of examples and motivation. For more details read .
with 
-classes 
in 
, then for any 
and 
, the subsets
{
} 
and 
{
} 
are called, respectively, cosets of A in B and cosets of B in A. These cosets partition B and A with
and 
. Cosets are always rectangular subalgebras in their 
-classes. What is more, the partial order 
induces a coset bijection 
defined by:
iff 
, for 
and 
.
Collectively, coset bijections describe
between the subsets 
and 
. They also determine 
and 
for pairs of elements from distinct 
-classes. Indeed, given 
and 
, let 
be the
cost bijection between the cosets
in 
and 
in 
. Then:
and 
.
In general, given
and 
with 
and 
, then 
belong to a common 
- coset in 
and 
belong to a common 
-coset in 
if and only if 
. Thus each coset bijection is, in some sense, a maximal collection of mutually parallel pairs 
.
Every primitive skew lattice
factors as the fibred product of its maximal left and right- handed primitive images 
. Right-handed primitive skew lattices are constructed as follows. Let 
and 
be partitions of disjoint nonempty sets 
and 
, where all 
and 
share a common size. For each pair 
pick a fixed bijection 
from 
onto 
. On 
and 
separately set 
and 
; but given 
and 
, set
and 
where
and 
with 
belonging to the cell 
of 
and 
belonging to the cell 
of 
. The various 
are the coset bijections. This is illustrated in the following partial Hasse diagram where 
and the arrows indicate the 
-outputs and 
from 
and 
.
One constructs left-handed primitive skew lattices in dual fashion. All right [left] handed primitive skew lattices can be constructed in this fashion. (See Section 1.)
is covered by its maximal primitive skew lattices: given comparable 
-classes 
in 
, 
forms a maximal primitive subalgebra of 
and every 
-class in 
lies in such a subalgebra. The coset structures on these primitive subalgebras combine to determine the outcomes 
and 
at least when 
and 
are comparable under 
. It turns out that 
and 
are determined in general by cosets and their bijections, although in
a slightly less direct manner than the
-comparable case. In particular, given two incomparable D-classes A and B with join D-class J and meet D-class 
in 
, interesting connections arise between the two coset decompositions of J (or M) with respect to A and B. (See Section 3.)
Thus a skew lattice may be viewed as a coset atlas of rectangular skew lattices placed on the vertices of a lattice and coset bijections between them, the latter seen as partial isomorphisms
between the rectangular algebras with each coset bijection determining a corresponding pair of cosets. This perspective gives, in essence, the Hasse diagram of the skew lattice, which is easily
drawn in cases of relatively small order. (See the diagrams in Section 3 above.) Given a chain of D-classes
in 
, one has three sets of coset bijections: from A to B, from B to C and from A to C. In general, given coset bijections 
and 
, the composition of partial bijections 
could be empty. If it is not, then a unique coset bijection 
exists such that 
. (Again, 
is a bijection between a pair of cosets in 
and 
.) This inclusion can be strict. It is always an equality (given 
) on a given skew lattice S precisely when S is categorical. In this case, by including the identity maps on each rectangular D-class and adjoining empty bijections between properly comparable D-classes, one has a category of rectangular algebras and coset bijections between them. The simple examples in Section 3 are categorical.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a skew lattice is an algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
that is a non-commutative generalization of a lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
. While the term skew lattice can be used to refer to any non-commutative generalization of a lattice, over the past twenty years it has been used primarily as follows.
Definition
A skew lattice is a set S equipped with two associative, idempotent binary operationBinary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
s
and , called meet and join, that satisfy the following dual pair of absorption laws and .Given that
and are associative and idempotent, these identities are equivalent to the dualities: iff and iff . Historical background
For over 60 years, noncommutative variations of lattices have been studied with differing motivations. For some the motivation has been an interest in the conceptual boundaries of lattice theory; for others it was a search for noncommutative forms of logicMathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
and Boolean algebra; and for others it has been the behavior of idempotents in rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
. A noncommutative lattice, generally speaking, is an algebra
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
where and are associative, idempotent binaryBinary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
operations
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
connected by absorption identities guaranteeing that
in some way dualizes . The precise identities chosen depends upon the underlying motivation, with differing choices producing distinct varieties of algebrasVariety (universal algebra)
In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic...
. Pascual Jordan, motivated by questions in quantum logic
Quantum logic
In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...
, initiated a study of noncommutative lattices in his 1949 paper, Über Nichtkommutative Verbande, choosing the absorption identities
He referred to those algebras satisfying them as Schrägverbände. By varying or augmenting these identities, Jordan and others obtained a number of varieties of noncommutative lattices.
Beginning with Jonathan Leech's 1989 paper, Skew lattices in rings, skew lattices as defined above have been the primary objects of study. This was aided by previous results about bands. This was especially the case for many of the basic properties.
Basic properties
Natural partial order and natural quasiorderIn a skew lattice
, the natural partial order is defined by if , or dually, . The natural preorderPreorder
In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
on
is given by if or dually . While and agree on lattices, properly refines in the noncommutative case. The induced natural equivalenceEquivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
is defined by if , that is, and
or dually, and . The blocks of the partition arelattice ordered by
iff and exist such that . This permits us to write Hasse diagramHasse diagram
In order theory, a branch of mathematics, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction...
s of skew lattices such as the following pair:
E.g., in the diagram on the left above, that
and are related is expressed by the dashedsegment. The slanted lines reveal the natural partial order between elements of the distinct
-classes. The elements , and form the singleton -classes.Rectangular Skew Lattices
Skew lattices consisting of a single
-class are called rectangular. They are characterized by the equivalent identities: , and . Rectangular skew lattices are isomorphic to skew lattices having the following construction (and conversely): given nonemptysets
and , on define and . The -class partition of a skew lattice , as indicated in the above diagrams, is the unique partition of into its maximal rectangular subalgebras, Moreover, is a congruence with the induced quotient algebra being the maximal lattice image of , thus making every skew lattice a lattice of rectangular subalgebras. This is the Clifford-McLean Theorem for skew lattices, first given for bands separately by Clifford and McLean. It is also known as the First Decomposition Theorem for skew lattices.Right (left) handed skew lattices and the Kimura factorization
A skew lattice is right-handed if it satisfies the identity
or dually, .These identities essentially assert that
and in each -class. Every skew lattice has a unique maximal right-handed image where the congruence is defined by if both and (or dually, and ). Likewise a skew lattice is left-handed if and in each -class. Again the maximal left-handed image of a skew lattice is the image where the congruence is defined in dual fashion to . Many examples of skew lattices are either right or left-handed. In the lattice of congruences, and is the identity congruence . The induced epimorphism factors through both induced epimorphisms and . Setting , the homomorphism defined by , induces an isomorphism . This is the Kimura factorization of into a fibred product of its maximal right and left-handed images.Like the Clifford-McLean Theorem, Kimura factorization (or the Second Decomposition Theorem for skew lattices) was first given for regular bands (that satisfy the middle absorption
identity,
). Indeed both and are regular band operations. The above symbols , and come, of course, from basic semigroup theory.For more details on the basic properties of a skew lattice please read and .
Subvarieties of skew lattices
Skew lattices form a variety. Rectangular skew lattices, left-handed and right-handed skew lattices all form subvarieties that are central to the basic structure theory of skew lattices. Here are severalmore.
Symmetric Skew Lattices
A skew lattice S is symmetric if for any
, iff . Occurrences of commutation are thus unambiguous for such skew lattices, with subsets of pairwise commuting elements generating commutative subalgebras, i.e, sublattices. ( This is not true for skew lattices in general.) Equational bases for this subvariety, first given by Spinks are: and .A lattice section of a skew lattice
is a sublattice of meeting each -class of at a single element. is thus an internal copy of the lattice with the composition being an isomorphism. All symmetric skew lattices for which |S/D| \leq \aleph_0 , admit a lattice section. Symmetric or not, having a lattice section guarantees that also has internal copies of and given respectively by and , where and are the and congruence classes of in . Thus and are isomorphisms (See ). This leads to a commuting diagram of embedding dualizing the preceding Kimura diagram.Cancellative Skew Lattices
A skew lattice is cancellative if
and implies and likewise and implies . Cancellatice skew lattices are symmetric and can be shown to form a variety. Unlike lattices, they need not be distributive, and conversely.Distributive Skew Lattices
Distributive skew lattices are determined by the identities:
(D1 ) (D’1 )Unlike lattices, (D1 ) and (D‘1 ) are not equivalent in general for skew lattices, but they are for symmetric skew lattices. (See,,.) The condition (D1 ) can be strengthened to
(D2 ) in which case (D‘1 ) is a consequence. A skew lattice satisfies both (D2) and its dual, , if and only if it factors as the product of a distributive lattice and a rectangular skew lattice. In this latter case (D2 ) can be strengthened to and . (D3 )On its own, (D3 ) is equivalent to (D2 ) when symmetry is added. (See.) We thus have six subvarieties of skew lattices determined respectively by (D1), (D2), (D3) and their duals.
Normal Skew Lattices
As seen above,
and satisfy the identity . Bands satisfying the stronger identity, , are called normal. A skew lattice is normal skew if it satisfiesFor each element a in a normal skew lattice
, the set defined by { } or equivalently {} is a sublattice of , and conversely. (Thus normal skew lattices have also been called local lattices.) When both and are normal, splits isomorphically into a product of a lattice and a rectangular skew lattice , and conversely. Thus both normal skew lattices and split skew lattices form varieties. Returning to distribution, so that characterizes the variety of distributive, normal skew lattices, and (D3) characterizes the variety of symmetric, distributive, normal skew lattices.Categorical Skew Lattices
A skew lattice is categorical if nonempty composites of coset bijections are coset bijections. Categorical skew lattices form a variety. Skew lattices in rings and normal skew lattices are examples
of algebras on this variety. Let
with , and , be the coset bijection from to taking to , be the coset bijection from to taking to and finally be the coset bijection from to taking to . A skew lattice is categorical if one always has the equality , ie. , if thecomposite partial bijection
if nonempty is a coset bijection from a -coset of to an -cosetof
. That is .All distributive skew lattices are categorical. Though symmetric skew lattices might not be. In a sense they reveal the independence between the properties of symmetry and distributivity.
For more details on these and other subvarieties of skew lattices please read and .
Skew boolean algebras
A zero element in a skew lattice S is an element 0 of S such that for all, or, dually, . (0)A Boolean skew lattice is a symmetric, distributive normal skew lattice with 0,
, such that is a Boolean lattice for each . Given such skew lattice S , a difference operator \ is defined on by x\ y = where the latter is evaluated in the Boolean lattice . In the presence of (D3) and (0), \ is characterized by the identities: and One thus has a variety of skew Boolean algebras
characterized by identities (D3), (0) and (S B). A primitive skew Boolean algebra consists of 0 and a single non-0 D-class. Thus it is the result of adjoining a 0 to a rectangular skew lattice D via (0) with , if and
otherwise. Every skew Boolean algebra is a subdirect product of primitive algebras. Skew Boolean algebras play an important role in the study of discriminator varieties and other generalizations in universal algebra of Boolean behavior. For more details on skew Boolean algebras see,,,,, ,,, and.Skew lattices in rings
Let be a RingRing (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
and let
denote the set of all Idempotents in . For all set and .Clearly
but also is associative. If a subset is closed under and , then is a distributive, cancellative skew lattice. To find such skew lattices in one looks at bands in , especially the ones that are maximal with respect to some constraint. In fact, every multiplicative band in that is maximal with respect to being right regular (= ) is also closed under and so forms a right-handed skew lattice. In general, every right regular band in generates a right-handed skew lattice in . Dual remarks also hold for left regular bands (bands satisfying the identity ) in . Maximal regular bands need not to be closed under as defined; counterexamples are easily found using multiplicative rectangular bands. These cases are closed, however, under the cubic variant of defined by since in these cases reduces to to give the dual rectangular band. By replacing the condition of regularity by normality , every maximal normal multiplicative band in is also closed under with , where , forms a Boolean skew lattice. When itself is closed under multiplication, then it is a normal band and thus forms a Boolean skew lattice. In fact, any skew Boolean algebra can be embedded into such an algebra. (See.) When A has a multiplicative identity , the condition that is multiplicatively closed is well-known to imply that forms a Boolean algebra. Skew lattices in rings continue to be a good source of examples and motivation. For more details read .Primitive skew lattices
Skew lattices consisting of exactly two D-classes are called primitive skew lattices. Given such a skew lattice with -classes in , then for any and , the subsets{} and {} are called, respectively, cosets of A in B and cosets of B in A. These cosets partition B and A with
and . Cosets are always rectangular subalgebras in their -classes. What is more, the partial order induces a coset bijection defined by: iff , for and .Collectively, coset bijections describe
between the subsets and . They also determine and for pairs of elements from distinct -classes. Indeed, given and , let be thecost bijection between the cosets
in and in . Then: and .In general, given
and with and , then belong to a common - coset in and belong to a common -coset in if and only if . Thus each coset bijection is, in some sense, a maximal collection of mutually parallel pairs .Every primitive skew lattice
factors as the fibred product of its maximal left and right- handed primitive images . Right-handed primitive skew lattices are constructed as follows. Let and be partitions of disjoint nonempty sets and , where all and share a common size. For each pair pick a fixed bijection from onto . On and separately set and ; but given and , set and where
and with belonging to the cell of and belonging to the cell of . The various are the coset bijections. This is illustrated in the following partial Hasse diagram where and the arrows indicate the -outputs and from and .One constructs left-handed primitive skew lattices in dual fashion. All right [left] handed primitive skew lattices can be constructed in this fashion. (See Section 1.)
The coset structure of skew lattices
A nonrectangular skew lattice is covered by its maximal primitive skew lattices: given comparable -classes in , forms a maximal primitive subalgebra of and every -class in lies in such a subalgebra. The coset structures on these primitive subalgebras combine to determine the outcomes and at least when and are comparable under . It turns out that and are determined in general by cosets and their bijections, although ina slightly less direct manner than the
-comparable case. In particular, given two incomparable D-classes A and B with join D-class J and meet D-class in , interesting connections arise between the two coset decompositions of J (or M) with respect to A and B. (See Section 3.)Thus a skew lattice may be viewed as a coset atlas of rectangular skew lattices placed on the vertices of a lattice and coset bijections between them, the latter seen as partial isomorphisms
between the rectangular algebras with each coset bijection determining a corresponding pair of cosets. This perspective gives, in essence, the Hasse diagram of the skew lattice, which is easily
drawn in cases of relatively small order. (See the diagrams in Section 3 above.) Given a chain of D-classes
in , one has three sets of coset bijections: from A to B, from B to C and from A to C. In general, given coset bijections and , the composition of partial bijections could be empty. If it is not, then a unique coset bijection exists such that . (Again, is a bijection between a pair of cosets in and .) This inclusion can be strict. It is always an equality (given ) on a given skew lattice S precisely when S is categorical. In this case, by including the identity maps on each rectangular D-class and adjoining empty bijections between properly comparable D-classes, one has a category of rectangular algebras and coset bijections between them. The simple examples in Section 3 are categorical.