Page 90 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
thus a refined quasilattice. In fact it is the fibered product of its maximal skew lattice and skew*
lattice images. (Corollary 3.4.6.) As a consequence we have the following sublattice of the
lattice of quasilattice varieties:
Refined quasilattices
Regular paralattices
skewlattices skew *lattices
lattices
In general a refined quasilattice is partially regular in that L(∨) and R(∨) are congruences relative
to ∨, while L(∧) and R(∧) are congruences relative to ∧. Even if not fully regular, refined
paralattices behave very much like skew(*) lattices. This aspect is developed in some detail in
the discussion from Theorem 3.4.7 through Theorem 3.4.14. The latter asserts that each refined
quasilattice, if not isomorphic, is at least isotopic to some skew lattice. (The definition of
“isotopic” is given in the section). Indeed every refined quasilattice can be viewed as the result of
taking a left-handed skew lattice and mildly scrambling its ∨- and ∧-computational details. (See
the remarks after Theorem 3.4.14.) Thus here is a sense in which refined quasilattices are
roughly skew lattices.
In Section 3.5 we study the effects of various distributive identities. Theorem 3.5.1
asserts that a double band (S; ∨, ∧) satisfying both a∧(b ∨ c)∧a = (a∧b∧a) ∨ (a∧c∧a) and its dual
a∨(b ∧ c)∨a = (a∨b∨a) ∧ (a∨c∨a) is a quasilattice if and only if it a paralattice. Theorem 3.5.2
asserts that such a quasilattice [paralattice] satisfies both strengthened identities
a∧(b ∨ c)∧d = (a∧b∧d) ∨ (a∧c∧d) and its dual a∨(b ∧ c)∨d = (a∨b∨d) ∧ (a∨c∨d) if and only if it
factors as the product of a distributive lattice and an antilattice. Finally, any given quasilattice
[paralattice] factors as the product of a distributive lattice and a regular antilattice if and only if
the even stronger identities, a∧(b ∨ c) = (a∧b) ∨ (a∧c), (b ∨ c)∧d = (b∧d) ∨ (c∧d) and their duals,
hold.
The sixth and final section is rather “recreational” in nature. Extending Section 3.2, we
consider ways that magic squares, finite planes and other rectangular designs can be used to
create simple antilattices. Many classic magic squares, beginning with the classic Lo Shu,
provide examples of simple antilattices.
816 ⎡8 1 6 ⎤ ⎡1 2 3⎤
357 (∧) ⎢⎢3 5 ⎥ (∨) ⎢⎢4 5 6⎥⎥
492 ⎯⎯→ 7 ⎥
⎣⎢4 9 2⎥⎦ ⎢⎣7 8 9 ⎦⎥
Conditions for simplicity are given for various classes of antilattices derived from finite designs.
(See, e.g., Theorems 4.6.10 and 4.6.11.)
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thus a refined quasilattice. In fact it is the fibered product of its maximal skew lattice and skew*
lattice images. (Corollary 3.4.6.) As a consequence we have the following sublattice of the
lattice of quasilattice varieties:
Refined quasilattices
Regular paralattices
skewlattices skew *lattices
lattices
In general a refined quasilattice is partially regular in that L(∨) and R(∨) are congruences relative
to ∨, while L(∧) and R(∧) are congruences relative to ∧. Even if not fully regular, refined
paralattices behave very much like skew(*) lattices. This aspect is developed in some detail in
the discussion from Theorem 3.4.7 through Theorem 3.4.14. The latter asserts that each refined
quasilattice, if not isomorphic, is at least isotopic to some skew lattice. (The definition of
“isotopic” is given in the section). Indeed every refined quasilattice can be viewed as the result of
taking a left-handed skew lattice and mildly scrambling its ∨- and ∧-computational details. (See
the remarks after Theorem 3.4.14.) Thus here is a sense in which refined quasilattices are
roughly skew lattices.
In Section 3.5 we study the effects of various distributive identities. Theorem 3.5.1
asserts that a double band (S; ∨, ∧) satisfying both a∧(b ∨ c)∧a = (a∧b∧a) ∨ (a∧c∧a) and its dual
a∨(b ∧ c)∨a = (a∨b∨a) ∧ (a∨c∨a) is a quasilattice if and only if it a paralattice. Theorem 3.5.2
asserts that such a quasilattice [paralattice] satisfies both strengthened identities
a∧(b ∨ c)∧d = (a∧b∧d) ∨ (a∧c∧d) and its dual a∨(b ∧ c)∨d = (a∨b∨d) ∧ (a∨c∨d) if and only if it
factors as the product of a distributive lattice and an antilattice. Finally, any given quasilattice
[paralattice] factors as the product of a distributive lattice and a regular antilattice if and only if
the even stronger identities, a∧(b ∨ c) = (a∧b) ∨ (a∧c), (b ∨ c)∧d = (b∧d) ∨ (c∧d) and their duals,
hold.
The sixth and final section is rather “recreational” in nature. Extending Section 3.2, we
consider ways that magic squares, finite planes and other rectangular designs can be used to
create simple antilattices. Many classic magic squares, beginning with the classic Lo Shu,
provide examples of simple antilattices.
816 ⎡8 1 6 ⎤ ⎡1 2 3⎤
357 (∧) ⎢⎢3 5 ⎥ (∨) ⎢⎢4 5 6⎥⎥
492 ⎯⎯→ 7 ⎥
⎣⎢4 9 2⎥⎦ ⎢⎣7 8 9 ⎦⎥
Conditions for simplicity are given for various classes of antilattices derived from finite designs.
(See, e.g., Theorems 4.6.10 and 4.6.11.)
88
