Page 36 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 36
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
published in the 1960s and early 1970s, studied algebras satisfying B1, B3, C2 and C4 which
characterize (r, l)-flat quasilattices. In a sequence of papers beginning in the 1980s, Gh. Farcas
considered systems combining (l, l)-flatness with middle commutativity, a∨b∨c = a∨c∨b and
a∧b∧c = a∧c∧b, so that both operations were left normal.
Distributive identities
Just as the number of essentially distinct absorption identities proliferates in the absence
of commutativity, so do the number of essentially distinct distributive identities. To begin, a
noncommutative lattice is fully distributive if it satisfies the identities:
D1 : a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c). D1′ : a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c).
D2 : (a ∨ b) ∧ c = (a ∧ c) ∨ (b ∧ c). D2′ : (a ∧ b) ∨ c = (a ∨ c) ∧ (b ∨ c).
Unlike the case for lattices:
Theorem 1.3.9. For skew lattices D1 , D1′ , D2 and D2′ are mutually independent.
Proof. A skew lattice satisfying precisely D1 , D1′ and D2′ is given by the tables:
∨ab0 and ∧ab0
aaaa a a b 0.
bbbb bab0
0ab0 0000
That D2′ is not satisfied is seen by (a ∧ 0) ∨ b = b ≠ a = (a ∨ b) ∧ (0 ∨ b). Replacing (∨, ∧) by
(∨*, ∧*) or (∧, ∨) or both (∧*,∨*) gives examples of the other ways that a skew lattice can satisfy
just three of the four identities above. £
Being fully distributive is powerful. To see this, observe first that distributive lattices are
fully distributive as well as flat antilattices of any of the four types. Hence any direct product of
algebras from these five types is also fully distributive. We shall see in Corollary 3.5.3 that for
quasilattices (and hence skew lattices) this is all. Thus, any fully distributive quasilattice factors
as the product of a distributive lattice and up to four flat antilattices.
We next consider two other sets of distributive identities that while less powerful, have a
broader scope. A noncommutative lattice is bidistributive if it satisfies the slightly weaker pair of
identities:
D3 : a ∧ (b ∨ c) ∧ d = (a ∧ b∧ d) ∨ (a ∧ c ∧ d).
D3 ʹ: a ∨ (b ∧ c) ∨ d = (a ∨ b∨ d) ∧ (a ∨ c ∨ d).
Distributive lattices and arbitrary antilattices are bidistributive, as are their direct products. By
Theorem 3.5.2 below, this is all among quasilattices. (In this more general case the antilattice
factor need not be completely factored into a product of flat antilattices.)
34
published in the 1960s and early 1970s, studied algebras satisfying B1, B3, C2 and C4 which
characterize (r, l)-flat quasilattices. In a sequence of papers beginning in the 1980s, Gh. Farcas
considered systems combining (l, l)-flatness with middle commutativity, a∨b∨c = a∨c∨b and
a∧b∧c = a∧c∧b, so that both operations were left normal.
Distributive identities
Just as the number of essentially distinct absorption identities proliferates in the absence
of commutativity, so do the number of essentially distinct distributive identities. To begin, a
noncommutative lattice is fully distributive if it satisfies the identities:
D1 : a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c). D1′ : a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c).
D2 : (a ∨ b) ∧ c = (a ∧ c) ∨ (b ∧ c). D2′ : (a ∧ b) ∨ c = (a ∨ c) ∧ (b ∨ c).
Unlike the case for lattices:
Theorem 1.3.9. For skew lattices D1 , D1′ , D2 and D2′ are mutually independent.
Proof. A skew lattice satisfying precisely D1 , D1′ and D2′ is given by the tables:
∨ab0 and ∧ab0
aaaa a a b 0.
bbbb bab0
0ab0 0000
That D2′ is not satisfied is seen by (a ∧ 0) ∨ b = b ≠ a = (a ∨ b) ∧ (0 ∨ b). Replacing (∨, ∧) by
(∨*, ∧*) or (∧, ∨) or both (∧*,∨*) gives examples of the other ways that a skew lattice can satisfy
just three of the four identities above. £
Being fully distributive is powerful. To see this, observe first that distributive lattices are
fully distributive as well as flat antilattices of any of the four types. Hence any direct product of
algebras from these five types is also fully distributive. We shall see in Corollary 3.5.3 that for
quasilattices (and hence skew lattices) this is all. Thus, any fully distributive quasilattice factors
as the product of a distributive lattice and up to four flat antilattices.
We next consider two other sets of distributive identities that while less powerful, have a
broader scope. A noncommutative lattice is bidistributive if it satisfies the slightly weaker pair of
identities:
D3 : a ∧ (b ∨ c) ∧ d = (a ∧ b∧ d) ∨ (a ∧ c ∧ d).
D3 ʹ: a ∨ (b ∧ c) ∨ d = (a ∨ b∨ d) ∧ (a ∨ c ∨ d).
Distributive lattices and arbitrary antilattices are bidistributive, as are their direct products. By
Theorem 3.5.2 below, this is all among quasilattices. (In this more general case the antilattice
factor need not be completely factored into a product of flat antilattices.)
34
