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, iff . 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 |S/D| \leq \aleph_0, 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 (See ). 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. (See,,.) 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. (See.) 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 on 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, ie., 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.
For more details on these and other subvarieties of skew lattices please read and.
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