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 formula_1 and formula_2, called "meet" and "join", that validate the following dual pair of absorption laws
formula_3,
formula_4.
Given that formula_5 and formula_6 are associative and idempotent, these identities are equivalent to validating the following dual pair of statements:
formula_7 if formula_8,
formula_9 if formula_10. 
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 formula_11 where formula_6 and formula_5 are associative, idempotent binary operations connected by absorption identities guaranteeing that formula_6 in some way dualizes formula_5. 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
formula_16
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 formula_17, the natural partial order is defined by formula_18 if formula_19, or dually, formula_20. The natural preorder on formula_17 is given by formula_22 if formula_23 or dually formula_24. While formula_25 and formula_26 agree on lattices, formula_25 properly refines formula_26 in the noncommutative case. The induced natural equivalence formula_29 is defined by formula_30 if formula_31, that is, formula_32
and formula_33 or dually, formula_24 and formula_35. The blocks of the partition formula_36 are
lattice ordered by formula_37 if formula_38 and formula_39 exist such that formula_40. 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 formula_41 and formula_42 are formula_29 related is expressed by the dashed
segment. The slanted lines reveal the natural partial order between elements of the distinct formula_29-classes. The elements formula_45, formula_46 and formula_47 form the singleton formula_29-classes.
Rectangular Skew Lattices
Skew lattices consisting of a single formula_29-class are called rectangular. They are characterized by the equivalent identities: formula_50, formula_35 and formula_52. Rectangular skew lattices are isomorphic to skew lattices having the following construction (and conversely): given nonempty
sets formula_53 and formula_54, on formula_55 define formula_56 and formula_57. The formula_29-class partition of a skew lattice formula_17, as indicated in the above diagrams, is the unique partition of formula_17 into its maximal rectangular subalgebras, Moreover, formula_29 is a congruence with the induced quotient algebra formula_62 being the maximal lattice image of formula_17 , thus making every skew lattice formula_17 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 formula_65 or dually, formula_66.
These identities essentially assert that formula_67 and formula_68 in each formula_29-class. Every skew lattice formula_17 has a unique maximal right-handed image formula_71 where the congruence formula_53 is defined by formula_73 if both formula_74 and formula_75 (or dually, formula_76 and formula_77). Likewise a skew lattice is left-handed if formula_74 and formula_76 in each formula_29-class. Again the maximal left-handed image of a skew lattice formula_17 is the image formula_82 where the congruence formula_54 is defined in dual fashion to formula_53. Many examples of skew lattices are either right or left-handed. In the lattice of congruences, formula_85 and formula_86 is the identity congruence formula_87. The induced epimorphism formula_88 factors through both induced epimorphisms formula_89 and formula_90. Setting formula_91, the homomorphism formula_92 defined by formula_93, induces an isomorphism formula_94. This is the Kimura factorization of formula_17 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, formula_96 ). Indeed, both formula_6 and formula_5 are regular band operations. The above symbols formula_29, formula_54 and formula_53 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 formula_102 , formula_103 if formula_104. 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:
formula_105 and formula_106.
A lattice section of a skew lattice formula_17 is a sublattice formula_108 of formula_17 meeting each formula_29-class of formula_17 at a single element. formula_108 is thus an internal copy of the lattice formula_62 with the composition formula_114 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 formula_108 guarantees that formula_17 also has internal copies of formula_71 and formula_82 given respectively by formula_119 and formula_120, where formula_121 and formula_122 are the formula_54 and formula_53 congruence classes of formula_125 in formula_108 . Thus formula_127 and formula_128 are isomorphisms. This leads to a commuting diagram of embedding dualizing the preceding Kimura diagram.
Cancellative Skew Lattices
A skew lattice is cancellative if formula_129 and formula_130 implies formula_131 and likewise formula_132 and formula_133 implies formula_134. 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:
formula_135 (D1)
formula_136 (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 formula_137 (D2) in which case (D'1) is a consequence. A skew lattice formula_17 satisfies both (D2) and its dual, formula_139, 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 formula_140 and formula_141. (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, formula_6 and formula_5 satisfy the identity formula_96. Bands satisfying the stronger identity, formula_145, are called normal. A skew lattice is normal skew if it satisfies
formula_146
For each element a in a normal skew lattice formula_17, the set formula_148 defined by { formula_149} or equivalently {formula_150} is a sublattice of formula_17, and conversely. (Thus normal skew lattices have also been called local lattices.) When both formula_1 and formula_2 are normal, formula_17 splits isomorphically into a product formula_155 of a lattice formula_108 and a rectangular skew lattice formula_29, and conversely. Thus both normal skew lattices and split skew lattices form varieties. Returning to distribution, formula_158 so that formula_159 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 formula_160 with formula_38, formula_39 and formula_163, formula_164 be the coset bijection from formula_165 to formula_166 taking formula_41 to formula_42, formula_169 be the coset bijection from formula_166 to formula_171 taking formula_42 to formula_46 and finally formula_174 be the coset bijection from formula_165 to formula_171 taking formula_41 to formula_46. A skew lattice formula_17 is categorical if one always has the equality formula_180, i.e. , if the
composite partial bijection formula_181 if nonempty is a coset bijection from a formula_171 -coset of formula_165 to an formula_165-coset
of formula_171 . That is formula_186.
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 formula_187 formula_188 or, dually, formula_189 (0)
A Boolean skew lattice is a symmetric, distributive normal skew lattice with 0, formula_190 such that formula_148 is a Boolean lattice for each formula_192 Given such skew lattice S, a difference operator \ is defined by x\ y = formula_193 where the latter is evaluated in the Boolean lattice formula_194 In the presence of (D3) and (0), \ is characterized by the identities:
formula_195 and formula_196 (S B)
One thus has a variety of skew Boolean algebras formula_197 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 formula_198, if formula_199
and formula_47 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 formula_165 be a ring and let formula_202 denote the set of all Idempotents in formula_165. For all formula_204 set formula_205 and formula_206.
Clearly formula_6 but also formula_5 is associative. If a subset formula_209 is closed under formula_6 and formula_5, then formula_212 is a distributive, cancellative skew lattice. To find such skew lattices in formula_202 one looks at bands in formula_202, especially the ones that are maximal with respect to some constraint. In fact, every multiplicative band in formula_215 that is maximal with respect to being right regular (= ) is also closed under formula_5 and so forms a right-handed skew lattice. In general, every right regular band in formula_202 generates a right-handed skew lattice in formula_202. Dual remarks also hold for left regular bands (bands satisfying the identity formula_219) in formula_202. Maximal regular bands need not to be closed under formula_5 as defined; counterexamples are easily found using multiplicative rectangular bands. These cases are closed, however, under the cubic variant of formula_5 defined by formula_223 since in these cases formula_224 reduces to formula_225 to give the dual rectangular band. By replacing the condition of regularity by normality formula_226, every maximal normal multiplicative band formula_17 in formula_202 is also closed under formula_229 with formula_230, where formula_231, forms a Boolean skew lattice. When formula_202 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 formula_45, the condition that formula_202 is multiplicatively closed is well known to imply that formula_202 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 formula_17 with formula_29-classes formula_37 in formula_36, then for any formula_38 and formula_39, the subsets
formula_242{formula_243} formula_244 and formula_245 {formula_246} formula_247
are called, respectively, "cosets of A in B" and "cosets of B in A". These cosets partition B and A with formula_248 and formula_249. Cosets are always rectangular subalgebras in their formula_29-classes. What is more, the partial order formula_251 induces a coset bijection formula_252 defined by:
formula_253 iff formula_254, for formula_255 and formula_256.
Collectively, coset bijections describe formula_251 between the subsets formula_165 and formula_166. They also determine formula_5 and formula_1 for pairs of elements from distinct formula_29-classes. Indeed, given formula_38 and formula_39, let formula_265 be the
cost bijection between the cosets formula_266 in formula_165 and formula_268 in formula_166. Then:
formula_270 and formula_271.
In general, given formula_272 and formula_273 with formula_40 and formula_275, then formula_276 belong to a common formula_166- coset in formula_165 and formula_279 belong to a common formula_165-coset in formula_166 if and only if formula_282. Thus each coset bijection is, in some sense, a maximal collection of mutually parallel pairs formula_40.
Every primitive skew lattice formula_17 factors as the fibred product of its maximal left and right- handed primitive images formula_285. Right-handed primitive skew lattices are constructed as follows. Let formula_286 and formula_287 be partitions of disjoint nonempty sets formula_165 and formula_166, where all formula_290 and formula_291 share a common size. For each pair formula_292 pick a fixed bijection formula_293 from formula_290 onto formula_291. On formula_165 and formula_166 separately set formula_298 and formula_299; but given formula_38 and formula_39, set
formula_302 and formula_303
where formula_304 and formula_305 with formula_306 belonging to the cell formula_290 of formula_41 and formula_309 belonging to the cell formula_291 of formula_42. The various formula_312 are the coset bijections. This is illustrated in the following partial Hasse diagram where formula_313 and the arrows indicate the formula_314 -outputs and formula_251 from formula_165 and formula_166.
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 formula_17 is covered by its maximal primitive skew lattices: given comparable formula_29-classes formula_37 in formula_36, formula_322 forms a maximal primitive subalgebra of formula_17 and every formula_29-class in formula_17 lies in such a subalgebra. The coset structures on these primitive subalgebras combine to determine the outcomes formula_326 and formula_327 at least when formula_328 and formula_329 are comparable under formula_330. It turns out that formula_326 and formula_327 are determined in general by cosets and their bijections, although in
a slightly less direct manner than the formula_330-comparable case. In particular, given two incomparable D-classes A and B with join D-class J and meet D-class formula_334 in formula_36, 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 formula_336 in formula_36, 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 formula_338 and formula_339, the composition of partial bijections formula_340 could be empty. If it is not, then a unique coset bijection formula_341 exists such that formula_342. (Again, formula_343 is a bijection between a pair of cosets in formula_165 and formula_171.) This inclusion can be strict. It is always an equality (given formula_346) 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.