Skew lattice


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, since 1989 it has been used primarily as follows.

Definition

A skew lattice is a set S equipped with two associative, idempotent binary operations and, called meet and join, that validate the following dual pair of absorption laws
,

.

Given that and are associative and idempotent, these identities are equivalent to validating the following dual pair of statements:
if,

if.

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 logic 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 Verbände, 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 quasiorder
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 if and exist such that. This permits us to draw Hasse diagrams 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 : 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 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 . 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 was first given for regular bands. Indeed, both and are regular band operations. The above symbols, and come, of course, from basic semigroup theory.

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 several
more.
Symmetric Skew Lattices
A skew lattice S is symmetric if for any , if. Occurrences of commutation are thus unambiguous for such skew lattices, with subsets of pairwise commuting elements generating commutative subalgebras, i.e., sublattices. 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. 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:


Unlike lattices, and are not equivalent in general for skew lattices, but they are for symmetric skew lattices. The condition can be strengthened to in which case is a consequence. A skew lattice satisfies both 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 can be strengthened to and.
On its own, is equivalent to when symmetry is added. We thus have six subvarieties of skew lattices determined respectively by,, 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. 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 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, i.e., 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.

Skew Boolean algebras

A zero element in a skew lattice S is an element 0 of S such that for all or, dually,
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 by x\ y = where the latter is evaluated in the Boolean lattice In the presence of and, \ is characterized by the identities:
and
One thus has a variety of skew Boolean algebras characterized by identities, and. 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 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.

Skew lattices in rings

Let 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 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. 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.

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 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 handed primitive skew lattices can be constructed in this fashion.

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 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 with respect to A and B.

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. 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. This inclusion can be strict. It is always an equality 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.