Page 188 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 188
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
A primitive skew lattice A > B is order-closed if for a, aʹ ∈ A and b, bʹ ∈ B, a, aʹ > b and
a > b, bʹ imply aʹ > bʹ.
a − − a′
?
?
b − − b′
A primitive skew lattice A > B is simply order-closed if a > b holds for all a ∈ A and all b ∈ B.
In this case the cosets of A and B in each other are precisely the singleton subsets. The following
characterization of order-closed primitive skew lattices is easily verified: a primitive skew lattice
is order-closed if and only if it factors as the product T × D of a simply order-closed primitive
skew lattice T with a rectangular skew lattice D.
A skew lattice is order-closed if all its primitive subalgebras are order-closed. Order-
closed skew lattices form a subvariety of skew lattices. Using variables x, y, u and v, with
w = x∧y∧u∧v∧x∧y,
one has the following generic situation between two D-classes ( denoting >):
x ∧ y – – – w ∨ (y ∧ x) ∨ w
.
w – – – x ∧ y ∧ v ∧ u ∧ x ∧ y
Thus [w∨(y∧x)∨w] ∧ (x∧y∧v∧u∧x∧y) ∧ [w∨(y∧x)∨w] = x∧y∧v∧u∧x∧y characterizes these skew
lattices and we have:
Proposition 5.4.12. Order-closed skew lattices are a variety that includes both normal
skew lattices and conormal skew lattices.
Proof. That we have a variety is clear. That it includes all normal skew lattices is due to the fact
that given a primitive normal skew lattice A > B, for any a ∈ A only one b ∈ B exists such that
a > b, thus making A > B satisfy the defining condition for order-closure in a trivial manner. £
The variety join of the varieties of normal skew lattices and conormal skew lattices is
thus included in the intersection of the varieties of strictly categorical and order-closed skew
lattices. Moreover, primitive skew lattices not satisfying the order-closed criterion above are
easily designed. See, e.g., the example of Theorem 2.4.3. Thus strictly categorical skew lattices
are not the join of the normal and conormal skew lattice varieties.
186
A primitive skew lattice A > B is order-closed if for a, aʹ ∈ A and b, bʹ ∈ B, a, aʹ > b and
a > b, bʹ imply aʹ > bʹ.
a − − a′
?
?
b − − b′
A primitive skew lattice A > B is simply order-closed if a > b holds for all a ∈ A and all b ∈ B.
In this case the cosets of A and B in each other are precisely the singleton subsets. The following
characterization of order-closed primitive skew lattices is easily verified: a primitive skew lattice
is order-closed if and only if it factors as the product T × D of a simply order-closed primitive
skew lattice T with a rectangular skew lattice D.
A skew lattice is order-closed if all its primitive subalgebras are order-closed. Order-
closed skew lattices form a subvariety of skew lattices. Using variables x, y, u and v, with
w = x∧y∧u∧v∧x∧y,
one has the following generic situation between two D-classes ( denoting >):
x ∧ y – – – w ∨ (y ∧ x) ∨ w
.
w – – – x ∧ y ∧ v ∧ u ∧ x ∧ y
Thus [w∨(y∧x)∨w] ∧ (x∧y∧v∧u∧x∧y) ∧ [w∨(y∧x)∨w] = x∧y∧v∧u∧x∧y characterizes these skew
lattices and we have:
Proposition 5.4.12. Order-closed skew lattices are a variety that includes both normal
skew lattices and conormal skew lattices.
Proof. That we have a variety is clear. That it includes all normal skew lattices is due to the fact
that given a primitive normal skew lattice A > B, for any a ∈ A only one b ∈ B exists such that
a > b, thus making A > B satisfy the defining condition for order-closure in a trivial manner. £
The variety join of the varieties of normal skew lattices and conormal skew lattices is
thus included in the intersection of the varieties of strictly categorical and order-closed skew
lattices. Moreover, primitive skew lattices not satisfying the order-closed criterion above are
easily designed. See, e.g., the example of Theorem 2.4.3. Thus strictly categorical skew lattices
are not the join of the normal and conormal skew lattice varieties.
186
