Page 95 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 95
III: Quasilattices, Paralattices and their Congruences
Not all quasilattices Q = ∪λDλ for which Q/D ≅ Λ arise from some such pair (Φ, Ψ) of
ascending and descending morphisms. Those that do are characterized by both (Q, ∨) and (Q, ∧)
being normal. This is just the double assertion of Theorem 1.2.16.
The above construction is relevant to the question: Given a band B with multiplication ∧,
is (B, ∧) the ∧-reduct of some quasilattice (B, ∨, ∧)?
If (B, ∨, ∧) exists, it is called a quasilattice closure of B. It is not unique, unless B is a
semilattice so that (B, ∨, ∧) is a lattice. Clearly, it is necessary that the maximal semilattice
image (B/D, ∧) form a lattice in that given any pair of D-classes M and N in B, a D-class J exists
which is the supremum of M and N in (B/D, ∧). This condition is also sufficient. Indeed, assume
that B has join classes. On each D-class M define a rectangular join ∨M. (Perhaps, let x∨M y = x.)
Using an ascending family Φ of ∨-morphisms between the M, create an operation ∨ on all of B
that yields a quasilattice, (B, ∨, ∧).
Proposition 3.1.8. A band B is a ∧-reduct of a quasilattice if and only if its maximal
semilattice image B/D is the ∧-reduct of a lattice.
Returning to quasilattice congruences, note that while the interval [D, ∇] is distributive,
in general Con(N) need not satisfy any identity beyond those common to all lattices. Indeed
[Δ, D] need only satisfy identities common to congruence lattices of antilattices. But consider
the antilattice defined on a given set N by x∧y = x = x∨y. In this case Con(N) is just the lattice
Equ(N) of all equivalences on N. But such lattices collectively satisfy only lattice identities
common to all lattices. We shall see in the next section that not all antilattices are so behaved.
Consider the following questions. Given quasilattice Q, when is Con(Q) distributive?
Even if Con(Q) is not distributive, under what conditions will instances of distribution occur? In
response we present three assertions, the first two following immediately from Theorem 3.1.1 and
the complete embedding D*: [Δ, D] in ∏iCon(Di) where the Di are all maximal antilattices in Q.
1. Con(Q) is distributive precisely when [Δ, D] is distributive. The latter occurs when
each D-class of Q has a distributive congruence lattice. (For example, this is the case
if all D-classes are simple as algebras.)
2. Given particular quasilattice congruences η and θi for i ∈ I, η ∧ supi(θi) = supi(η∧θi)
holds in Con(Q) if either η or at least one of the θi lies in [D, ∇].
We present our third assertion as a theorem where we consider the remaining Green’s
equivalences L(∨), R(∨), L(∧) and R(∧) which are defined for any noncommutative lattice.
93
Not all quasilattices Q = ∪λDλ for which Q/D ≅ Λ arise from some such pair (Φ, Ψ) of
ascending and descending morphisms. Those that do are characterized by both (Q, ∨) and (Q, ∧)
being normal. This is just the double assertion of Theorem 1.2.16.
The above construction is relevant to the question: Given a band B with multiplication ∧,
is (B, ∧) the ∧-reduct of some quasilattice (B, ∨, ∧)?
If (B, ∨, ∧) exists, it is called a quasilattice closure of B. It is not unique, unless B is a
semilattice so that (B, ∨, ∧) is a lattice. Clearly, it is necessary that the maximal semilattice
image (B/D, ∧) form a lattice in that given any pair of D-classes M and N in B, a D-class J exists
which is the supremum of M and N in (B/D, ∧). This condition is also sufficient. Indeed, assume
that B has join classes. On each D-class M define a rectangular join ∨M. (Perhaps, let x∨M y = x.)
Using an ascending family Φ of ∨-morphisms between the M, create an operation ∨ on all of B
that yields a quasilattice, (B, ∨, ∧).
Proposition 3.1.8. A band B is a ∧-reduct of a quasilattice if and only if its maximal
semilattice image B/D is the ∧-reduct of a lattice.
Returning to quasilattice congruences, note that while the interval [D, ∇] is distributive,
in general Con(N) need not satisfy any identity beyond those common to all lattices. Indeed
[Δ, D] need only satisfy identities common to congruence lattices of antilattices. But consider
the antilattice defined on a given set N by x∧y = x = x∨y. In this case Con(N) is just the lattice
Equ(N) of all equivalences on N. But such lattices collectively satisfy only lattice identities
common to all lattices. We shall see in the next section that not all antilattices are so behaved.
Consider the following questions. Given quasilattice Q, when is Con(Q) distributive?
Even if Con(Q) is not distributive, under what conditions will instances of distribution occur? In
response we present three assertions, the first two following immediately from Theorem 3.1.1 and
the complete embedding D*: [Δ, D] in ∏iCon(Di) where the Di are all maximal antilattices in Q.
1. Con(Q) is distributive precisely when [Δ, D] is distributive. The latter occurs when
each D-class of Q has a distributive congruence lattice. (For example, this is the case
if all D-classes are simple as algebras.)
2. Given particular quasilattice congruences η and θi for i ∈ I, η ∧ supi(θi) = supi(η∧θi)
holds in Con(Q) if either η or at least one of the θi lies in [D, ∇].
We present our third assertion as a theorem where we consider the remaining Green’s
equivalences L(∨), R(∨), L(∧) and R(∧) which are defined for any noncommutative lattice.
93
