Page 35 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 35
I: Preliminaries
Thus, every quasilattice is a lattice of antilattices. Hence quasilattices are precisely the
noncommutative lattices having a direct analogue of the Clifford-McLean Theorem for bands.
Flatness
A noncommutative lattice is flat if one of the following holds:
(r, l): a∨b∨a = b∨a and a∧b∧a = a∧b.
(l, r): a∨b∨a = a∨b and a∧b∧a = b∧a.
(l, l): a∨b∨a = a∨b and a∧b∧a = a∧b.
(r, r): a∨b∨a = b∨a and a∧b∧a = b∧a.
Thus, being (r, l)-flat means that D(∨) = R(∨) and D(∧) = L(∧), or equivalently, L(∨) = R(∧) = Δ.
Modified remarks hold for the other three types of flatness. Clearly:
Proposition 1.3.7. Given any variety of noncommutative lattices (e.g., paralattices or
quasilattices), the flat algebras of a given type form a subvariety. £
If (N, ∨, ∧) if flat, then (N, ∨*, ∧), (N, ∨, ∧*) and (N, ∨*, ∧*) are also flat, with all four
types of flatness represented. Of particular interest is:
Theorem 1.3.8. (l, l)-flat paralattices are characterized by B1, B4, C1 and C4:
a∧(a∨b) = a = (a∨b)∧a and a∨(a∧b) = a = (a∧b)∨a.
Similarly, (l, l)-flat quasilattices are characterized by B1, B3, C1 and C3:
a∧(a∨b) = a = a∧(b∨a) and a∨(a∧b) = a = a∨(b∧a).
Proof. Assuming (l, l)-flatness B6, B7, C6 and C7 reduce to B1, B4, C1 and C4. Conversely, from
B1, B4, C1 and C4 we get a∧b∧a = a∧b∧[(a∧b)∨a]= a∧b. Switching ∨ and ∧ (which we can since
B1, B4, C1 and C4 are operational duals) yields a∨b∨a = a∨b. Thus B6, B7, C6 and C7 can be
recovered from B1, B4, C1 and C4. Still assuming (l, l)-flatness, B5, C5 and their derived
identities, a∧(a∨b∨a)∧a = a = a∨(a∧b∧a)∨a, reduce to B1, B3, C1 and C3. From the latter,
a∧b∧a = a∧b∧[a∨(a∧b)] = a∧b follows. Similarly, a∨b∨a = a∨b. Thus B5 and C5 are recovered
from B1, B3, C1 and C3. £
This illustrates a general rule of thumb: upon assuming flatness, B1 - B4 and C1 - C4
usually suffice to describe the dualities encountered in noncommutative lattice theory.
Flatness was encountered in early work on noncommutative lattices. P. Jordan, who
wrote numerous articles on the subject in the 1950s and 1960s, often worked with algebras
satisfying B2, B3, C1 and C4 which characterize (r, l)-flat paralattices. M. D. Gerhardts, who
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Thus, every quasilattice is a lattice of antilattices. Hence quasilattices are precisely the
noncommutative lattices having a direct analogue of the Clifford-McLean Theorem for bands.
Flatness
A noncommutative lattice is flat if one of the following holds:
(r, l): a∨b∨a = b∨a and a∧b∧a = a∧b.
(l, r): a∨b∨a = a∨b and a∧b∧a = b∧a.
(l, l): a∨b∨a = a∨b and a∧b∧a = a∧b.
(r, r): a∨b∨a = b∨a and a∧b∧a = b∧a.
Thus, being (r, l)-flat means that D(∨) = R(∨) and D(∧) = L(∧), or equivalently, L(∨) = R(∧) = Δ.
Modified remarks hold for the other three types of flatness. Clearly:
Proposition 1.3.7. Given any variety of noncommutative lattices (e.g., paralattices or
quasilattices), the flat algebras of a given type form a subvariety. £
If (N, ∨, ∧) if flat, then (N, ∨*, ∧), (N, ∨, ∧*) and (N, ∨*, ∧*) are also flat, with all four
types of flatness represented. Of particular interest is:
Theorem 1.3.8. (l, l)-flat paralattices are characterized by B1, B4, C1 and C4:
a∧(a∨b) = a = (a∨b)∧a and a∨(a∧b) = a = (a∧b)∨a.
Similarly, (l, l)-flat quasilattices are characterized by B1, B3, C1 and C3:
a∧(a∨b) = a = a∧(b∨a) and a∨(a∧b) = a = a∨(b∧a).
Proof. Assuming (l, l)-flatness B6, B7, C6 and C7 reduce to B1, B4, C1 and C4. Conversely, from
B1, B4, C1 and C4 we get a∧b∧a = a∧b∧[(a∧b)∨a]= a∧b. Switching ∨ and ∧ (which we can since
B1, B4, C1 and C4 are operational duals) yields a∨b∨a = a∨b. Thus B6, B7, C6 and C7 can be
recovered from B1, B4, C1 and C4. Still assuming (l, l)-flatness, B5, C5 and their derived
identities, a∧(a∨b∨a)∧a = a = a∨(a∧b∧a)∨a, reduce to B1, B3, C1 and C3. From the latter,
a∧b∧a = a∧b∧[a∨(a∧b)] = a∧b follows. Similarly, a∨b∨a = a∨b. Thus B5 and C5 are recovered
from B1, B3, C1 and C3. £
This illustrates a general rule of thumb: upon assuming flatness, B1 - B4 and C1 - C4
usually suffice to describe the dualities encountered in noncommutative lattice theory.
Flatness was encountered in early work on noncommutative lattices. P. Jordan, who
wrote numerous articles on the subject in the 1950s and 1960s, often worked with algebras
satisfying B2, B3, C1 and C4 which characterize (r, l)-flat paralattices. M. D. Gerhardts, who
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