Page 99 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 99
III: Quasilattices, Paralattices and their Congruences
Theorem 3.3.1. The condition that L(∨) be a congruence determines a subvariety for
any variety of noncommutative lattices. It contains as a further subvariety those algebras for
which L(∨) = Δ. When L(∨) is a congruence on a noncommutative lattice N, then L(∨) = Δ on N/θ
for any congruence θ in [L(∨), ∇]. Similar remarks hold for D(∨), R(∨), D(∧), R(∧) and L(∧).
Proof. A typical pair of L(∨)-related elements are b∨a and a∨b∨a. L(∨) is compatible with ∨ and
∧ if and only if the equations,
[c∨( b∨a)]∨[c∨(a∨b∨a)] = c∨( b∨a) and [c∨(a∨b∨a)]∨[c∨(b∨a)] = c∨(a∨b∨a)
together with their -∨c, c∧- and -∧c variants, hold. Clearly L(∨) is a congruence when L(∨) = Δ, a
condition that is equivalent to the equation a∨b∨a = b∨a being satisfied. The first assertion
follows. If L(∨) is a congruence, then L(∨) = Δ on N/L(∨). Thus, if θ is as stated, N/θ, being a
homomorphic image of N/L(∨), must also satisfy L(∨) = Δ.
The verifications of the remaining cases are similar, depending on the descriptions of
typical R(∨)-related elements (a∨b∨a, and a∨b), typical D(∨)-related elements (a∨b∨a and
b∨a∨b) and their ∧-variants. £
Theorem 3.3.2. Flat quasilattices are regular. Regular quasilattices, in turn, form the
subvariety of quasilattices generated from all flat quasilattices.
Proof. In a flat quasilattice, each of R(∨), L(∨), R(∧) and L(∧) is either D or Δ, and hence is a
congruence. Conversely, given a regular quasilattice Q, define κ: Q → Q/L(∨) × Q/R(∨) by
κ(x) = (xL(∨), xR(∨)). Since L(∨)∩R(∨) = Δ, κ is an embedding of Q into a product of quasilattices
for which either D = R(∨) or D = L(∨) holds on each factor. Repeating the process on each factor,
but using R(∧) and L(∧) instead, we obtain a final embedding of Q into a product of four flat
quasilattices, each representing a specific type of flatness. The theorem follows. £
Given a noncommutative lattice Q, let ϕ(l, l) denote the least congruence on θ for which
Q/θ is (l, l)-flat. Similarly, let ϕ(l, r), ϕ(r, l) and ϕ(r, r) denote least congruences for the other types
of flatness. For regular quasilattices, Theorems 3.1.8 and 4.3.1 yield:
Lemma 3.3.3. Given a regular quasilattice Q, the equivalences R(∨), L(∨), R(∧) and L(∧)
all permute with each other. Moreover, ϕ(l, l) = R(∨)oR(∧), ϕ(l, r) = R(∨)oL(∧), ϕ(r, l) = L(∨)oR(∧)
and ϕ(r, r) = L(∨)oL(∧). £
97
Theorem 3.3.1. The condition that L(∨) be a congruence determines a subvariety for
any variety of noncommutative lattices. It contains as a further subvariety those algebras for
which L(∨) = Δ. When L(∨) is a congruence on a noncommutative lattice N, then L(∨) = Δ on N/θ
for any congruence θ in [L(∨), ∇]. Similar remarks hold for D(∨), R(∨), D(∧), R(∧) and L(∧).
Proof. A typical pair of L(∨)-related elements are b∨a and a∨b∨a. L(∨) is compatible with ∨ and
∧ if and only if the equations,
[c∨( b∨a)]∨[c∨(a∨b∨a)] = c∨( b∨a) and [c∨(a∨b∨a)]∨[c∨(b∨a)] = c∨(a∨b∨a)
together with their -∨c, c∧- and -∧c variants, hold. Clearly L(∨) is a congruence when L(∨) = Δ, a
condition that is equivalent to the equation a∨b∨a = b∨a being satisfied. The first assertion
follows. If L(∨) is a congruence, then L(∨) = Δ on N/L(∨). Thus, if θ is as stated, N/θ, being a
homomorphic image of N/L(∨), must also satisfy L(∨) = Δ.
The verifications of the remaining cases are similar, depending on the descriptions of
typical R(∨)-related elements (a∨b∨a, and a∨b), typical D(∨)-related elements (a∨b∨a and
b∨a∨b) and their ∧-variants. £
Theorem 3.3.2. Flat quasilattices are regular. Regular quasilattices, in turn, form the
subvariety of quasilattices generated from all flat quasilattices.
Proof. In a flat quasilattice, each of R(∨), L(∨), R(∧) and L(∧) is either D or Δ, and hence is a
congruence. Conversely, given a regular quasilattice Q, define κ: Q → Q/L(∨) × Q/R(∨) by
κ(x) = (xL(∨), xR(∨)). Since L(∨)∩R(∨) = Δ, κ is an embedding of Q into a product of quasilattices
for which either D = R(∨) or D = L(∨) holds on each factor. Repeating the process on each factor,
but using R(∧) and L(∧) instead, we obtain a final embedding of Q into a product of four flat
quasilattices, each representing a specific type of flatness. The theorem follows. £
Given a noncommutative lattice Q, let ϕ(l, l) denote the least congruence on θ for which
Q/θ is (l, l)-flat. Similarly, let ϕ(l, r), ϕ(r, l) and ϕ(r, r) denote least congruences for the other types
of flatness. For regular quasilattices, Theorems 3.1.8 and 4.3.1 yield:
Lemma 3.3.3. Given a regular quasilattice Q, the equivalences R(∨), L(∨), R(∧) and L(∧)
all permute with each other. Moreover, ϕ(l, l) = R(∨)oR(∧), ϕ(l, r) = R(∨)oL(∧), ϕ(r, l) = L(∨)oR(∧)
and ϕ(r, r) = L(∨)oL(∧). £
97
