Page 44 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 44
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
exhibits a nonsymmetric 13-element skew lattice with a generating set of 3 mutually commuting
elements (under both ∨ and ∧), that has noncommuting pairs of elements.
In the third section we consider normal skew lattices, i.e., skew lattices whose ∧-reducts
are normal in that x∧y∧z∧w = x∧y∧z∧w. Of special interest are distributive, symmetric, normal
skew lattices characterized in Theorem 2.3.2 by identities a∧(b ∨ c) = (a∧b) ∨ (a∧c) and
(a ∨ b)∧c = (a∧c) ∨ (b∧c). This strengthened form of distributivity is called strong distributivity.
Thanks to Theorem 2.3.6, every normal skew lattice of idempotents in a ring is strongly
distributive. In this case the operations are given by e∧f = ef again, but
e∨f = (e +f – ef)2 = e +f +fe – efe – fef.
Of course when e +f – ef is idempotent, both outcomes agree. Strongly distributive skew lattices
are also of interest due to their connections to skew Boolean algebras, the subject of Chapter 4.
Suffice it to say here that a skew lattice can be embedded into (the skew lattice reduct of) a skew
Boolean algebra precisely when it is strongly distributive.
In Section 4 we engage in a detailed study of the natural partial order ≥ on a skew lattice.
This study is based on the behavior of primitive skew lattices consisting of exactly two D-classes,
A > B. Primitive skew lattices have a simple description given in terms of A-cosets arising in B
and B-cosets arising in A and the coset bijections between these cosets induced by ≥. (See
Theorem 2.4.1) As a consequence, primitive skew lattices of the most general type are easily
manufactured. Moreover, the interaction between the various maximal primitive subalgebras of a
skew lattice says much about the behavior of the entire algebra. (See, e.g., Theorem 2.4.9 and its
consequences.) We also look at skew chains of comparable D-classes A > B > C and consider
the case where coset bijections between cosets in the outer classes are compositions of successive
intermediate coset bijections (as is the case for skew lattices in rings). When this always occurs
in a skew lattice, it is said to be categorical.
In Section 5 we continue the analysis of skew lattices by their primitive subalgebras
begun in the previous section. But here we pass from coset bijections between comparable pairs
of cosets in (usually) distinct D-classes to coset projections from an entire D-class onto a coset in
a comparable D-class. The individual projections are obtained by combining all coset bijections
from one D-class that share a common coset of outputs in the other class. Even if the skew
lattice itself is not categorical in the above sense, these downward (or upward) projections taken
collectively along with the involved D-classes, form a category. (See Theorem 2.5.4.) All this is
developed in the fifth section and then applied to give a general description of a normal skew
lattice in Theorem 2.5.7.
In Section 6 we study decompositions of (mostly symmetric) normal skew lattices. After
some preliminary cases, theorems of general character are given. For instance, the Reduction
Theorem (2.6.9) implies that every symmetric normal skew lattice can be embedded in
the product of its maximal lattice image and its maximal distributive image. The Primary
Decomposition Theorem (2.6.11) tells how a strongly distributive (hence symmetric and normal)
42
exhibits a nonsymmetric 13-element skew lattice with a generating set of 3 mutually commuting
elements (under both ∨ and ∧), that has noncommuting pairs of elements.
In the third section we consider normal skew lattices, i.e., skew lattices whose ∧-reducts
are normal in that x∧y∧z∧w = x∧y∧z∧w. Of special interest are distributive, symmetric, normal
skew lattices characterized in Theorem 2.3.2 by identities a∧(b ∨ c) = (a∧b) ∨ (a∧c) and
(a ∨ b)∧c = (a∧c) ∨ (b∧c). This strengthened form of distributivity is called strong distributivity.
Thanks to Theorem 2.3.6, every normal skew lattice of idempotents in a ring is strongly
distributive. In this case the operations are given by e∧f = ef again, but
e∨f = (e +f – ef)2 = e +f +fe – efe – fef.
Of course when e +f – ef is idempotent, both outcomes agree. Strongly distributive skew lattices
are also of interest due to their connections to skew Boolean algebras, the subject of Chapter 4.
Suffice it to say here that a skew lattice can be embedded into (the skew lattice reduct of) a skew
Boolean algebra precisely when it is strongly distributive.
In Section 4 we engage in a detailed study of the natural partial order ≥ on a skew lattice.
This study is based on the behavior of primitive skew lattices consisting of exactly two D-classes,
A > B. Primitive skew lattices have a simple description given in terms of A-cosets arising in B
and B-cosets arising in A and the coset bijections between these cosets induced by ≥. (See
Theorem 2.4.1) As a consequence, primitive skew lattices of the most general type are easily
manufactured. Moreover, the interaction between the various maximal primitive subalgebras of a
skew lattice says much about the behavior of the entire algebra. (See, e.g., Theorem 2.4.9 and its
consequences.) We also look at skew chains of comparable D-classes A > B > C and consider
the case where coset bijections between cosets in the outer classes are compositions of successive
intermediate coset bijections (as is the case for skew lattices in rings). When this always occurs
in a skew lattice, it is said to be categorical.
In Section 5 we continue the analysis of skew lattices by their primitive subalgebras
begun in the previous section. But here we pass from coset bijections between comparable pairs
of cosets in (usually) distinct D-classes to coset projections from an entire D-class onto a coset in
a comparable D-class. The individual projections are obtained by combining all coset bijections
from one D-class that share a common coset of outputs in the other class. Even if the skew
lattice itself is not categorical in the above sense, these downward (or upward) projections taken
collectively along with the involved D-classes, form a category. (See Theorem 2.5.4.) All this is
developed in the fifth section and then applied to give a general description of a normal skew
lattice in Theorem 2.5.7.
In Section 6 we study decompositions of (mostly symmetric) normal skew lattices. After
some preliminary cases, theorems of general character are given. For instance, the Reduction
Theorem (2.6.9) implies that every symmetric normal skew lattice can be embedded in
the product of its maximal lattice image and its maximal distributive image. The Primary
Decomposition Theorem (2.6.11) tells how a strongly distributive (hence symmetric and normal)
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