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 and only if ,
if and only 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,Jordan, P. Uber Nichtkommutative Verbände, Arch. Math. 2 (1949), 56–59. 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,Leech, J, Skew lattices in rings, Algebra Universalis, 26(1989), 48-72 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 only 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.Leech, J, Recent developments in the theory of skew lattices, Semigroup Forum, 52(1996), 7-24.Leech, J, Magic squares, finite planes and simple quasilattices, Ars Combinatoria 77(2005), 75-96.Leech, J, The geometry of skew lattices, Semigroup Forum, 52(1993), 7-24.Leech, J, Normal skew lattices, Semigroup Forum, 44(1992), 1-8.Cvetko-Vah, K, Internal decompositions of skew lattices, Communications in Algebra, 35 (2007), 243-247Cvetko-Vah, K, A new proof of Spinks’ Theorem, Semigroup Forum 73 (2006), 267-272.Laslo, G and Leech, J, Green’s relations on noncommutative lattices, Acta Sci. Math. (Szeged), 68 (2002), 501-533.
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 and only 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 SpinksSpinks, M, Automated deduction in non-commutative lattice theory, Tech. Report 3/98, Monash U, GSCIT, 1998 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.Spinks, M, Automated deduction in non-commutative lattice theory, Tech. Report 3/98, Monash University, Gippsland School of Computing and Information Technology, June 1998 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.Spinks, M, On middle distributivity for skew lattices, Semigroup Forum 61 (2000), 341-345.Cvetko-Vah, Karin ; Kinyon, M. ; Leech, J. ; Spinks, M. Cancellation in skew Lattices. Order 28 (2011), 9-32.
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.Bignall, R. J., Quasiprimal Varieties and Components of Universal Algebras, Dissertation, The Flinders University of South Australia, 1976.Bignall, R J, A non-commutative multiple-valued logic, Proc. 21st International Symposium on Multiple-valued Logic, 1991, IEEE Computer Soc. Press, 49-54.Bignall, R J and J Leech, Skew Boolean algebras and discriminator varieties, Algebra Universalis, 33(1995), 387-398.Bignall, R J and M Spinks, Propositional skew Boolean logic, Proc. 26th International Symposium on Multiple-valued Logic, 1996, IEEE Computer Soc. Press, 43-48.Bignall, R J and M Spinks, Implicative BCS-algebra subreducts of skew Boolean algebras, Scientiae Mathematicae Japonicae, 58 (2003), 629-638.Bignall, R J and M Spinks, On binary discriminator varieties (I): Implicative BCS-algebras, International Journal of Algebra and Computation, to appear.Cornish, W H, Boolean skew algebras, Acta Math. Acad. Sci. Hung., 36 (1980), 281-291.Leech, J, Skew Boolean algebras, Algebra Universalis, 27(1990), 497-506.Leech and Spinks, Skew Boolean algebras generated from generalized Boolean algebras, Algebra Universalis 58 (2008), 287-302, 307-311.Spinks, M, Contributions to the Theory of Pre-BCK Algebras, Monash University Dissertation, 2002.Spinks, M and R Veroff, Axiomatizing the skew Boolean propositional calculus, J. Automated Reasoning, 37 (2006), 3-20.
Skew lattices in rings Let be a ring and let denote the set of all idempotents in . For all set and .