Page 189 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 189
V: Further Topics in Skew Lattices
A skew lattice is paranormal if is it is an order-closed, strictly categorical skew lattice.
We can now pose the following question:
Is the variety of paranormal skew lattices jointly generated by the varieties of normal and
conormal skew lattices? Otherwise put, is it their join in the lattice of all skew lattice varieties?
One can also consider conditioned versions of the question by asking if, say, the variety
of symmetric paranormal skew lattices is the join of the varieties of symmetric normal and
symmetric conormal skew lattices. “Symmetric,” of course, could be replaced by “distributive”
or by “distributive and symmetric.” In the latter case, a related question is:
Is the variety of distributive, symmetric, paranormal skew lattices generated from the
class of all order-closed primitive algebras?
The motivation for this is the fact that distributive, symmetric, normal skew lattices are
generated from 3R and 3L. (See Theorem 2.6.12.) Indeed, returning to strictly categorical skew
lattices in general, one may ask:
Is the variety of distributive, symmetric, strictly categorical skew lattices generated from
the class of all primitive algebras?
5.5 Distributive skew lattices
We begin with a broader class of skew lattices. A skew lattice S is linearly distributive if
every subalgebra T that is totally preordered under ≻ is distributive. Since totally preordered
skew lattices are trivially symmetric, a skew lattice S is linearly distributive if and only if each
totally preordered subalgebra satisfies either (5.2.1) or equivalently (5.2.2). Indeed, S is linearly
distributive if it is distributive on each skew chain A ≥ B ≥ C in S. Since skew chains need not be
even categorical, they need not be distributive! However:
Theorem 5.5.1. Linearly distributive skew lattices form a variety of skew lattices. Thus
a skew lattice S is linearly distributive if and only if both S/R and S/L are.
Proof. Consider the terms x, y∧x∧y and z∧y∧x∧y∧z. Clearly x ≻ y∧x∧y ≻ z∧y∧x∧y∧z holds for
all skew lattices. Conversely given any instance a ≻ b ≻ c in some skew lattice S, the assignment
x → a, y → b, z → c will return this particular instance. Thus a characterizing set of identities for
the class of all linearly distributive skew lattices is given by taking the basic identity
u∧(v ∨ w)∧u = (u∧v∧u) ∨ (u∧w∧u)
and forming all the identities possible in x, y, z by making bijective assignments from the
variables {u, v, w} to the terms {x, y∧x∧y, z∧y∧x∧y∧z}. £
187
A skew lattice is paranormal if is it is an order-closed, strictly categorical skew lattice.
We can now pose the following question:
Is the variety of paranormal skew lattices jointly generated by the varieties of normal and
conormal skew lattices? Otherwise put, is it their join in the lattice of all skew lattice varieties?
One can also consider conditioned versions of the question by asking if, say, the variety
of symmetric paranormal skew lattices is the join of the varieties of symmetric normal and
symmetric conormal skew lattices. “Symmetric,” of course, could be replaced by “distributive”
or by “distributive and symmetric.” In the latter case, a related question is:
Is the variety of distributive, symmetric, paranormal skew lattices generated from the
class of all order-closed primitive algebras?
The motivation for this is the fact that distributive, symmetric, normal skew lattices are
generated from 3R and 3L. (See Theorem 2.6.12.) Indeed, returning to strictly categorical skew
lattices in general, one may ask:
Is the variety of distributive, symmetric, strictly categorical skew lattices generated from
the class of all primitive algebras?
5.5 Distributive skew lattices
We begin with a broader class of skew lattices. A skew lattice S is linearly distributive if
every subalgebra T that is totally preordered under ≻ is distributive. Since totally preordered
skew lattices are trivially symmetric, a skew lattice S is linearly distributive if and only if each
totally preordered subalgebra satisfies either (5.2.1) or equivalently (5.2.2). Indeed, S is linearly
distributive if it is distributive on each skew chain A ≥ B ≥ C in S. Since skew chains need not be
even categorical, they need not be distributive! However:
Theorem 5.5.1. Linearly distributive skew lattices form a variety of skew lattices. Thus
a skew lattice S is linearly distributive if and only if both S/R and S/L are.
Proof. Consider the terms x, y∧x∧y and z∧y∧x∧y∧z. Clearly x ≻ y∧x∧y ≻ z∧y∧x∧y∧z holds for
all skew lattices. Conversely given any instance a ≻ b ≻ c in some skew lattice S, the assignment
x → a, y → b, z → c will return this particular instance. Thus a characterizing set of identities for
the class of all linearly distributive skew lattices is given by taking the basic identity
u∧(v ∨ w)∧u = (u∧v∧u) ∨ (u∧w∧u)
and forming all the identities possible in x, y, z by making bijective assignments from the
variables {u, v, w} to the terms {x, y∧x∧y, z∧y∧x∧y∧z}. £
187
