Skew Lattice - Skew Lattices in Rings

Skew Lattices in Rings

Let be a Ring and let denote the set of all Idempotents in . For all set and .

Clearly but also is associative. If a subset is closed under and, then is a distributive, cancellative skew lattice. To find such skew lattices in one looks at bands in, especially the ones that are maximal with respect to some constraint. In fact, every multiplicative band in that is maximal with respect to being right regular (= ) is also closed under and so forms a right-handed skew lattice. In general, every right regular band in generates a right-handed skew lattice in . Dual remarks also hold for left regular bands (bands satisfying the identity ) in . Maximal regular bands need not to be closed under as defined; counterexamples are easily found using multiplicative rectangular bands. These cases are closed, however, under the cubic variant of defined by since in these cases reduces to to give the dual rectangular band. By replacing the condition of regularity by normality, every maximal normal multiplicative band in is also closed under with, where, forms a Boolean skew lattice. When itself is closed under multiplication, then it is a normal band and thus forms a Boolean skew lattice. In fact, any skew Boolean algebra can be embedded into such an algebra. (See.) When A has a multiplicative identity, the condition that is multiplicatively closed is well-known to imply that forms a Boolean algebra. Skew lattices in rings continue to be a good source of examples and motivation. For more details read.