Page 31 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 31
Preliminaries
The noncommutative lattices studied most extensively in recent years are skew lattices
that satisfy B1, B2, C1 and C2. We will see that these identities express the dualities:
x ∨ y = x if and only if x ∧ y = y and x ∨ y = y if and only if x ∧ y = x.
(The equivalence of the absorption identities with the stated dualities will be shown shortly.)
Likewise, skew* lattices satisfy the complementary set of identities, B3, B4, C3 and C4, and are
characterized by the pair of dualities: x∨y = x if and only if y∧x = y and x∨y = y if and only if
y∧x = x. Either type of algebra is transformed into the other by replacing ∧ by ∧* or by replacing
∨ by ∨* where a ∧* b = b ∧ a and a ∨* b = b ∨ a. A double replacement yields the original type
of algebra. Hence we choose to focus on skew lattices. A detailed study of skew lattices will
commence in the following chapter. Two further classes of noncommutative lattices are:
A quasilattice is a noncommutative lattice satisfying the two-sided absorption identities
B5: a ∧ (b ∨ a ∨ b) ∧ a = a. C5: a ∨ (b ∧ a ∧ b) ∨ a = a.
We will see that these identities expresses the duality, x ∧ y ∧ x = x if and only if y ∨ x ∨ y = y,
stating that x ≺ y under ∧ if and only if y ≺ x under ∨.
A paralattice is a noncommutative lattice satisfying the absorption identities
B6: a ∧ (a ∨ b ∨ a) = a = (a ∨ b ∨ a) ∧ a.
C6: a ∨ (a ∧ b ∧ a) = a = (a ∧ b ∧ a) ∨ a.
These identities express the duality, x ∧ y = x = y ∧ x if and only if x ∨ y = y = y ∨ x, stating
that x ≤ y under ∧ if and only if y ≤ x under ∨.
While B5 – C6 above are not among the previous absorption identities, if combined with
flatness (see below), they reduce to identities on the earlier list. (See Theorem 1.3.7 below.)
Quasilattices that are also paralattices are called refined quasi-lattices. Skew lattices and
skew* lattices are both refined quasi-lattices. Another significant class of examples is as follows.
In Section 3.4 we will see that refined quasi-lattices are closely related to skew* lattices.
An antilattice is an algebra (N, ∨, ∧) with associative, idempotent binary operations ∨
and ∧ such that both a ∧ b ∧ a = a and a ∨ b ∨ a = a. Lattices and antilattices form antipodal
classes of examples having foundational import in the study of noncommutative lattices. Thanks
to the simple behavior of rectangular bands, however, antilattices are easily described. A pair of
double indexing {x(λ, ρ) ⎢(λ, ρ) ∈L × R} and {x(λ*,ρ*) ⎢(λ*, ρ*) ∈ L* × R*} of the elements of N
exist such that:
x(λ, ρ) ∨ x(µ, σ) = x(λ, σ) and x(λ*, ρ*) ∧ x(µ*, σ*) = x(λ*, σ*).
If N is finite, it can be exhibited as a pair of rectangular arrays. Consider the example:
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The noncommutative lattices studied most extensively in recent years are skew lattices
that satisfy B1, B2, C1 and C2. We will see that these identities express the dualities:
x ∨ y = x if and only if x ∧ y = y and x ∨ y = y if and only if x ∧ y = x.
(The equivalence of the absorption identities with the stated dualities will be shown shortly.)
Likewise, skew* lattices satisfy the complementary set of identities, B3, B4, C3 and C4, and are
characterized by the pair of dualities: x∨y = x if and only if y∧x = y and x∨y = y if and only if
y∧x = x. Either type of algebra is transformed into the other by replacing ∧ by ∧* or by replacing
∨ by ∨* where a ∧* b = b ∧ a and a ∨* b = b ∨ a. A double replacement yields the original type
of algebra. Hence we choose to focus on skew lattices. A detailed study of skew lattices will
commence in the following chapter. Two further classes of noncommutative lattices are:
A quasilattice is a noncommutative lattice satisfying the two-sided absorption identities
B5: a ∧ (b ∨ a ∨ b) ∧ a = a. C5: a ∨ (b ∧ a ∧ b) ∨ a = a.
We will see that these identities expresses the duality, x ∧ y ∧ x = x if and only if y ∨ x ∨ y = y,
stating that x ≺ y under ∧ if and only if y ≺ x under ∨.
A paralattice is a noncommutative lattice satisfying the absorption identities
B6: a ∧ (a ∨ b ∨ a) = a = (a ∨ b ∨ a) ∧ a.
C6: a ∨ (a ∧ b ∧ a) = a = (a ∧ b ∧ a) ∨ a.
These identities express the duality, x ∧ y = x = y ∧ x if and only if x ∨ y = y = y ∨ x, stating
that x ≤ y under ∧ if and only if y ≤ x under ∨.
While B5 – C6 above are not among the previous absorption identities, if combined with
flatness (see below), they reduce to identities on the earlier list. (See Theorem 1.3.7 below.)
Quasilattices that are also paralattices are called refined quasi-lattices. Skew lattices and
skew* lattices are both refined quasi-lattices. Another significant class of examples is as follows.
In Section 3.4 we will see that refined quasi-lattices are closely related to skew* lattices.
An antilattice is an algebra (N, ∨, ∧) with associative, idempotent binary operations ∨
and ∧ such that both a ∧ b ∧ a = a and a ∨ b ∨ a = a. Lattices and antilattices form antipodal
classes of examples having foundational import in the study of noncommutative lattices. Thanks
to the simple behavior of rectangular bands, however, antilattices are easily described. A pair of
double indexing {x(λ, ρ) ⎢(λ, ρ) ∈L × R} and {x(λ*,ρ*) ⎢(λ*, ρ*) ∈ L* × R*} of the elements of N
exist such that:
x(λ, ρ) ∨ x(µ, σ) = x(λ, σ) and x(λ*, ρ*) ∧ x(µ*, σ*) = x(λ*, σ*).
If N is finite, it can be exhibited as a pair of rectangular arrays. Consider the example:
29
