Skew Lattice - 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 iff and exist such that . This permits us to write 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 (that satisfy the middle absorption identity, ). Indeed both and are regular band operations. The above symbols, and come, of course, from basic semigroup theory.

For more details on the basic properties of a skew lattice please read and.