Page 109 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 109
III: Quasilattices, Paralattices and their Congruences
y = z∨y = (x∧x∧z)∨(x∧y∧z) = x∧(x∨y)∧z = x∧x∧z = z.
Thus if x ≥ y,z with y L(∧) z, then y = z. If we had first supposed that y R(∧) z, then it would
follow in similar fashion that y = z. Thus x ≥ y,z in N with y D z implies that y = z, from which
the identity u∧x∧y∧u = u∧y∧x∧u follows.
Similarly, if x, y ≥ z in N with x D y, then D6 can be used to show that x = y, from which
follows the identity u∨x∨y∨u = u∨y∨x∨u. By Theorem 3.1.4, N factors into the product of its
(necessarily) distributive maximal lattice image and an antilattice.
For the converse, observe that antilattices are always bidistributive as, of course, are
distributive lattices. Thus split quasilattices whose lattice factors are distributive are indeed
bidistributive quasilattices. £
An even stronger result occurs when full distribution is assumed:
Corollary 3.5.3. A quasilattice is fully distributive if and only if it is the product of a
distributive lattice and a regular antilattice. In particular, every fully distributive quasilattice is
regular.
Proof. Let N be a fully distributive quasilattice. N factors as the product of a distributive lattice
T and an antilattice A. Being a fine quasilattice, L(∨) is a ∨-congruence by Theorem 3.4.5. But
L(∨) is a also a ∧-congruence: given b L(∨) c, D1 implies a∧b L(∨) a∧c and D2 implies
b∧d L(∨) c∧d. Thus L(∨) is a full congruence. Similarly, R(∨) is a full congruence. Likewise, D3
and D4 imply that R(∧) and L(∧) are full congruences. Thus both N and its rectangular factor A
are regular. Conversely, let N factor as the product of a distributive lattice T and a regular
antilattice A. Being regular, A factors into a product of flat algebras, A ≅ A(l,l) × A(l,r) × A(r,l) ×
A(r,r), with each factor having operations ∨ and ∧ that each satisfy ab = a or ab = b. Conversely,
identities D1 – D4 are easily seen to hold on such structures and, of course on T and hence on N.
£
As mentioned above, two lines of research meet at fully distributive quasilattices. In
particular, our Corollary 4.5.3 follows from Theorem 2.6 of Pastijn [1983]. By the latter theorem,
the lattice of varieties of all ID-semirings satisfying the identities,
x + y + z + w = x + z + y + w and xyzw = xzyw,
is described. The sublattice of those varieties that also satisfy semiring versions of B5 and C5 is
easily seen to be the join of the variety of distributive lattices and the variety of rectangular ID-
semirings (satisfying x + y + z = x + z and xyz = xz). Thus every fully distributive quasilattice is
the subdirect product of a distributive lattice and a rectangular ID-semiring (that is essentially a
regular rectangular quasilattice). But such a subdirect product splits by Theorem 3.5.2 and the
corollary follows.
107
y = z∨y = (x∧x∧z)∨(x∧y∧z) = x∧(x∨y)∧z = x∧x∧z = z.
Thus if x ≥ y,z with y L(∧) z, then y = z. If we had first supposed that y R(∧) z, then it would
follow in similar fashion that y = z. Thus x ≥ y,z in N with y D z implies that y = z, from which
the identity u∧x∧y∧u = u∧y∧x∧u follows.
Similarly, if x, y ≥ z in N with x D y, then D6 can be used to show that x = y, from which
follows the identity u∨x∨y∨u = u∨y∨x∨u. By Theorem 3.1.4, N factors into the product of its
(necessarily) distributive maximal lattice image and an antilattice.
For the converse, observe that antilattices are always bidistributive as, of course, are
distributive lattices. Thus split quasilattices whose lattice factors are distributive are indeed
bidistributive quasilattices. £
An even stronger result occurs when full distribution is assumed:
Corollary 3.5.3. A quasilattice is fully distributive if and only if it is the product of a
distributive lattice and a regular antilattice. In particular, every fully distributive quasilattice is
regular.
Proof. Let N be a fully distributive quasilattice. N factors as the product of a distributive lattice
T and an antilattice A. Being a fine quasilattice, L(∨) is a ∨-congruence by Theorem 3.4.5. But
L(∨) is a also a ∧-congruence: given b L(∨) c, D1 implies a∧b L(∨) a∧c and D2 implies
b∧d L(∨) c∧d. Thus L(∨) is a full congruence. Similarly, R(∨) is a full congruence. Likewise, D3
and D4 imply that R(∧) and L(∧) are full congruences. Thus both N and its rectangular factor A
are regular. Conversely, let N factor as the product of a distributive lattice T and a regular
antilattice A. Being regular, A factors into a product of flat algebras, A ≅ A(l,l) × A(l,r) × A(r,l) ×
A(r,r), with each factor having operations ∨ and ∧ that each satisfy ab = a or ab = b. Conversely,
identities D1 – D4 are easily seen to hold on such structures and, of course on T and hence on N.
£
As mentioned above, two lines of research meet at fully distributive quasilattices. In
particular, our Corollary 4.5.3 follows from Theorem 2.6 of Pastijn [1983]. By the latter theorem,
the lattice of varieties of all ID-semirings satisfying the identities,
x + y + z + w = x + z + y + w and xyzw = xzyw,
is described. The sublattice of those varieties that also satisfy semiring versions of B5 and C5 is
easily seen to be the join of the variety of distributive lattices and the variety of rectangular ID-
semirings (satisfying x + y + z = x + z and xyz = xz). Thus every fully distributive quasilattice is
the subdirect product of a distributive lattice and a rectangular ID-semiring (that is essentially a
regular rectangular quasilattice). But such a subdirect product splits by Theorem 3.5.2 and the
corollary follows.
107
