Skew lattice

A skew lattice is a set S equipped with two associative, idempotent binary operations

, called meet and join, that validate the following dual pair of absorption laws Given that

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

are associative, idempotent binary operations connected by absorption identities guaranteeing that

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,[2] 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.

The slanted lines reveal the natural partial order between elements of the distinct

This is the Clifford–McLean theorem for skew lattices, first given for bands separately by Clifford and McLean.

into a fibred product of its maximal right- and left-handed images.

Cancellatice skew lattices are symmetric and can be shown to form a variety.

[1] We thus have six subvarieties of skew lattices determined respectively by (D1), (D2), (D3) and their duals.

characterizes the variety of distributive, normal skew lattices, and (D3) characterizes the variety of symmetric, distributive, normal skew lattices.

In a sense they reveal the independence between the properties of symmetry and distributivity.

A primitive skew Boolean algebra consists of 0 and a single non-0 D-class.

as defined; counterexamples are easily found using multiplicative rectangular bands.

itself is closed under multiplication, then it is a normal band and thus forms a Boolean skew lattice.

Skew lattices in rings continue to be a good source of examples and motivation.

These cosets partition B and A with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.")

Thus each coset bijection is, in some sense, a maximal collection of mutually parallel pairs

factors as the fibred product of its maximal left and right- handed primitive images

One constructs left-handed primitive skew lattices in dual fashion.

All right [left] handed primitive skew lattices can be constructed in this fashion.

is covered by its maximal primitive skew lattices: given comparable

The coset structures on these primitive subalgebras combine to determine the outcomes

are determined in general by cosets and their bijections, although in a slightly less direct manner than the

, interesting connections arise between the two coset decompositions of J (or M) with respect to A and B.

This perspective gives, in essence, the Hasse diagram of the skew lattice, which is easily drawn in cases of relatively small order.

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.