Page 96 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Theorem 3.1.9. If Q is a quasilattice, then L(∨), R(∨), L(∧) and R(∧) permute with any
congruence in [Δ, D], ∧-distribute over suprema in [Δ, D] and ∨-distribute over infima in [Δ, D],
even if these four equivalences are not congruences themselves. If either L(∨), R(∨), L(∧) or R(∧)
is also a congruence, then it possesses these distributive properties in Con(Q).
Proof. To begin, L o θ = θ o L and R o θ = θ o R hold for any congruence θ on a rectangular
band. Hence L(∨), R(∨), L(∧) and R(∧) permute with any congruence on an antilattice. Likewise,
L and R distribute as stated over congruences on a rectangular band. Thus the first statement
must hold at least for antilattices. The general statement involving [Δ, D] now follows since all
calculations take place in D-classes. Since the join of all four equivalences with D is just D, the
second statement follows from the first and Theorem 3.1.2. £
3.2 Antilattices that are simple as algebras
Recall that an algebra A is simple if Con(A) = {Δ, ∇}. Algebras of order 2 are simple.
By Theorem 3.1.2, a simple quasilattice is either a simple lattice or a simple antilattice. Our
interest is in the latter case. We begin with some remarks about when simplicity cannot occur,
starting with a definition.
An antilattice N is flat if L(∨), R(∨), L(∧) and R(∧) are each either Δ or ∇. In the finite
case, N is flat precisely when the arrays defining ∨ and ∧ are either single columns or rows.
Lemma 3.2.1. All antilattices of prime order are flat; moreover every equivalence on a
flat antilattice is a congruence. Thus an antilattice cannot be simple if its order is an odd prime.
Proof. Given the assumption of prime order, L(∨) and R(∨) reduce to either Δ or ∇. Hence either
x∨y = x holds uniformly or else x∨y = y holds uniformly. Similarly, either x∧y = x holds
uniformly or else x∧y = y does. For such operations, every equivalence must be a congruence. £
By contrast, let A be the antilattice on {a, b, c, d} determined from the arrays:
(∨) a b (∧) a d
cd c b.
Recall that x∨y and x∧y are the element lying in the row of x and the column of y of the relevant
array, where the rows are the R-classes and the columns are the L-classes for the respective
operations. Here L(∨) = L(∧) is the only proper nontrivial congruence. The lattice of congruences
is thus: Δ < L < ∇. In general:
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Theorem 3.1.9. If Q is a quasilattice, then L(∨), R(∨), L(∧) and R(∧) permute with any
congruence in [Δ, D], ∧-distribute over suprema in [Δ, D] and ∨-distribute over infima in [Δ, D],
even if these four equivalences are not congruences themselves. If either L(∨), R(∨), L(∧) or R(∧)
is also a congruence, then it possesses these distributive properties in Con(Q).
Proof. To begin, L o θ = θ o L and R o θ = θ o R hold for any congruence θ on a rectangular
band. Hence L(∨), R(∨), L(∧) and R(∧) permute with any congruence on an antilattice. Likewise,
L and R distribute as stated over congruences on a rectangular band. Thus the first statement
must hold at least for antilattices. The general statement involving [Δ, D] now follows since all
calculations take place in D-classes. Since the join of all four equivalences with D is just D, the
second statement follows from the first and Theorem 3.1.2. £
3.2 Antilattices that are simple as algebras
Recall that an algebra A is simple if Con(A) = {Δ, ∇}. Algebras of order 2 are simple.
By Theorem 3.1.2, a simple quasilattice is either a simple lattice or a simple antilattice. Our
interest is in the latter case. We begin with some remarks about when simplicity cannot occur,
starting with a definition.
An antilattice N is flat if L(∨), R(∨), L(∧) and R(∧) are each either Δ or ∇. In the finite
case, N is flat precisely when the arrays defining ∨ and ∧ are either single columns or rows.
Lemma 3.2.1. All antilattices of prime order are flat; moreover every equivalence on a
flat antilattice is a congruence. Thus an antilattice cannot be simple if its order is an odd prime.
Proof. Given the assumption of prime order, L(∨) and R(∨) reduce to either Δ or ∇. Hence either
x∨y = x holds uniformly or else x∨y = y holds uniformly. Similarly, either x∧y = x holds
uniformly or else x∧y = y does. For such operations, every equivalence must be a congruence. £
By contrast, let A be the antilattice on {a, b, c, d} determined from the arrays:
(∨) a b (∧) a d
cd c b.
Recall that x∨y and x∧y are the element lying in the row of x and the column of y of the relevant
array, where the rows are the R-classes and the columns are the L-classes for the respective
operations. Here L(∨) = L(∧) is the only proper nontrivial congruence. The lattice of congruences
is thus: Δ < L < ∇. In general:
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