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 (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 (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. 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. (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  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:

  (D1)

 (D'1)

Unlike lattices, (D1) and (D'1) are not equivalent in general for skew lattices, but they are for symmetric skew lattices. 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. 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 in 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,  (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 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  (S B)

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. 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 (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. 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 [left] 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 (or M) 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. (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. See also
Semigroup theory
Lattice theory

References

Lattice theory
Semigroup theory