Skew Lattice - Primitive Skew Lattices

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. (See Section 1.)