Page 38 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Enriched structures
We have already encountered 1 and 0. As is the case for lattices 1, if it exists, is the
unique element at the top of the noncommutative lattice and 0, if it exists, is the unique element at
the bottom, whatever the (quasi-)ordering may be. This is given expression by the following sets
of identities.
1 ∨ a = 1 = a ∨ 1 and 1 ∧ a = a = a ∧ 1.
0 ∧ a = 0 = a ∧ 0 and 0 ∨ a = a = a ∨ 0.
For skew lattices (and for paralattices in general) both second pairs of identities is redundant.
Paralattices, and particularly skew lattices, have a coherent natural partial order ≤ given
as either ≤(∧) or the dual of ≤(∨). It may be that two elements x, y in a paralattice possesses a
natural meet that is maximal among all z such that both x, y ≥ z. Dually they may also possess a
natural join that is minimal among all z such that both x, y ≤ z. To distinguish natural meets and
natural joins from the given operations ∧ and ∨, we shall refer to them as intersections and
unions of elements respectively, employing the notation x∩y and x∪y. Unless x and y commute,
one has x∧y > x∩y and x∪y > x∨y.
Theorem 1.3.12. The natural meet operation in a paralattice is characterized by the
identities:
NM1: x∩x = x;
NM2: x∩y = y∩x;
NM3: (x ∩ y) ∩ z = x ∩ ( y∩ z);
NM4: x ∧ (x∩y) = x∩y = (x∩y) ∧ x;
NM5: x ∩ (x∧y∧x) = x∧y∧x.
Dual identities characterize a natural joins. £
Theorem 1.3.13. Given a paralattice (P; ∨, ∧) with a natural meet ∩, the enriched
algebra (P; ∨, ∧, ∩) has a distributive congruence lattice.
Proof. Setting m(x, y, z) = (x∩y)∨(y∩z)∨(x∩z), one has m(x, x, y) = m(x, y, x) = m(y, x, x) = x.
But this implies its congruence lattice is distributive. (Theorem II.12.3 in Burris and
Sankappanavar [1981].) £
Rectangular bands and rectangular skew lattices
A rectangular skew lattice is a skew lattice (S; ∧, ∨) for which (S; ∧) and (S; ∨) are
rectangular bands. Equivalently, its is an antilattice that is also a skew lattice. Given a
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Enriched structures
We have already encountered 1 and 0. As is the case for lattices 1, if it exists, is the
unique element at the top of the noncommutative lattice and 0, if it exists, is the unique element at
the bottom, whatever the (quasi-)ordering may be. This is given expression by the following sets
of identities.
1 ∨ a = 1 = a ∨ 1 and 1 ∧ a = a = a ∧ 1.
0 ∧ a = 0 = a ∧ 0 and 0 ∨ a = a = a ∨ 0.
For skew lattices (and for paralattices in general) both second pairs of identities is redundant.
Paralattices, and particularly skew lattices, have a coherent natural partial order ≤ given
as either ≤(∧) or the dual of ≤(∨). It may be that two elements x, y in a paralattice possesses a
natural meet that is maximal among all z such that both x, y ≥ z. Dually they may also possess a
natural join that is minimal among all z such that both x, y ≤ z. To distinguish natural meets and
natural joins from the given operations ∧ and ∨, we shall refer to them as intersections and
unions of elements respectively, employing the notation x∩y and x∪y. Unless x and y commute,
one has x∧y > x∩y and x∪y > x∨y.
Theorem 1.3.12. The natural meet operation in a paralattice is characterized by the
identities:
NM1: x∩x = x;
NM2: x∩y = y∩x;
NM3: (x ∩ y) ∩ z = x ∩ ( y∩ z);
NM4: x ∧ (x∩y) = x∩y = (x∩y) ∧ x;
NM5: x ∩ (x∧y∧x) = x∧y∧x.
Dual identities characterize a natural joins. £
Theorem 1.3.13. Given a paralattice (P; ∨, ∧) with a natural meet ∩, the enriched
algebra (P; ∨, ∧, ∩) has a distributive congruence lattice.
Proof. Setting m(x, y, z) = (x∩y)∨(y∩z)∨(x∩z), one has m(x, x, y) = m(x, y, x) = m(y, x, x) = x.
But this implies its congruence lattice is distributive. (Theorem II.12.3 in Burris and
Sankappanavar [1981].) £
Rectangular bands and rectangular skew lattices
A rectangular skew lattice is a skew lattice (S; ∧, ∨) for which (S; ∧) and (S; ∨) are
rectangular bands. Equivalently, its is an antilattice that is also a skew lattice. Given a
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