Page 102 - 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
aA1c ∨ aA2c ∨ … ∨ aAnc = a(A1 ∪ A2 ∪ … ∪ An)c.
(B1, ≤) is hence an upper semilattice. In similar fashion one handles the free left normal case
(with expressions aA) and the right normal case (with expressions Ac).
Indeed with a little more work one can show that if B is the free (left, right or 2-sided)
regular band on X (where xyxzx = xyzx), then B1 is a ∨-band. With even further work one can
show that if B is a free band on X, then B1 is a ∨-band. £
Dual ∨-bands are paralattices for which ∧ is commutative. Dual ∨-bands arise as reducts
(S; ∨, ∩) of structurally enriched skew lattices (S; ∨, ∧, ∩) for which the natural partial ordering
≤ has natural meets, with inf(x, y) denoted by x∩y. In particular, they arise in the study of certain
types of skew Boolean algebras. (See Section 4.2 below.) For ∨-bands and their duals (where
D = D(∨)) we have:
Theorem 3.4.3. D is a congruence on a [dual] ∨-band B if and only if D = Δ and B is a
lattice. In general a paralattice is also a quasilattice if and only if both D(∨) and D(∧) are
congruences. Regular paralattices in particular are quasilattices.
Proof. Given a D-class A of a ∨-band B, pick elements a, b in A. If D = D(∧) is a congruence,
then a∨b lies in A also with a ≤ a∨b and b ≤ a∨b. Being a common D-class, this forces
a = a∨b = b. Thus A is a singleton set. Hence D = Δ and B is a lattice. More generally, let P be a
paralattice for which both D(∨) and D(∧) are congruences. Then P/D(∨) is a ∨-band. Since D(∧)
is a congruence on P, D(∧) is also a congruence on P/D(∨) by Theorem 3.3.1. Hence P/D(∨) is a
lattice by our first assertion and λ ⊆ D(∨). Similarly, λ ⊆ D(∧). Since the converse inclusions
hold, D(∨) = D(∧) and P is a quasilattice by Corollary 1.3.5. Conversely, when P is also a
quasilattice, D(∨) = D(∧) with the common equivalence being a congruence. £
Corollary 3.4.4. The class of all regular paralattices is just the class of regular refined
quasilattices. In particular, every flat refined quasilattice is necessarily regular and is either a
skew lattice or a skew* lattice.
Proof. The first statement is clear. Next, take, e.g., a (l,l)-flat refined quasilattice, P. Since P is
a flat quasilattice, ∨≺L = ∨≺ dualizes ∧≺L = ∧≺. Since P is a flat paralattice, ∨≺R = ∨≤ dualizes
∧≺R = ∧≤. Hence N is a flat skew* lattice and thus necessarily regular. £
Theorem 3.4.5. The variety of regular paralattices is the join of the varieties of skew
lattices and skew* lattices. In particular, every regular paralattice factors as the fibred product
of its maximal skew lattice image with its maximal skew* lattice images over its maximal lattice
image.
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aA1c ∨ aA2c ∨ … ∨ aAnc = a(A1 ∪ A2 ∪ … ∪ An)c.
(B1, ≤) is hence an upper semilattice. In similar fashion one handles the free left normal case
(with expressions aA) and the right normal case (with expressions Ac).
Indeed with a little more work one can show that if B is the free (left, right or 2-sided)
regular band on X (where xyxzx = xyzx), then B1 is a ∨-band. With even further work one can
show that if B is a free band on X, then B1 is a ∨-band. £
Dual ∨-bands are paralattices for which ∧ is commutative. Dual ∨-bands arise as reducts
(S; ∨, ∩) of structurally enriched skew lattices (S; ∨, ∧, ∩) for which the natural partial ordering
≤ has natural meets, with inf(x, y) denoted by x∩y. In particular, they arise in the study of certain
types of skew Boolean algebras. (See Section 4.2 below.) For ∨-bands and their duals (where
D = D(∨)) we have:
Theorem 3.4.3. D is a congruence on a [dual] ∨-band B if and only if D = Δ and B is a
lattice. In general a paralattice is also a quasilattice if and only if both D(∨) and D(∧) are
congruences. Regular paralattices in particular are quasilattices.
Proof. Given a D-class A of a ∨-band B, pick elements a, b in A. If D = D(∧) is a congruence,
then a∨b lies in A also with a ≤ a∨b and b ≤ a∨b. Being a common D-class, this forces
a = a∨b = b. Thus A is a singleton set. Hence D = Δ and B is a lattice. More generally, let P be a
paralattice for which both D(∨) and D(∧) are congruences. Then P/D(∨) is a ∨-band. Since D(∧)
is a congruence on P, D(∧) is also a congruence on P/D(∨) by Theorem 3.3.1. Hence P/D(∨) is a
lattice by our first assertion and λ ⊆ D(∨). Similarly, λ ⊆ D(∧). Since the converse inclusions
hold, D(∨) = D(∧) and P is a quasilattice by Corollary 1.3.5. Conversely, when P is also a
quasilattice, D(∨) = D(∧) with the common equivalence being a congruence. £
Corollary 3.4.4. The class of all regular paralattices is just the class of regular refined
quasilattices. In particular, every flat refined quasilattice is necessarily regular and is either a
skew lattice or a skew* lattice.
Proof. The first statement is clear. Next, take, e.g., a (l,l)-flat refined quasilattice, P. Since P is
a flat quasilattice, ∨≺L = ∨≺ dualizes ∧≺L = ∧≺. Since P is a flat paralattice, ∨≺R = ∨≤ dualizes
∧≺R = ∧≤. Hence N is a flat skew* lattice and thus necessarily regular. £
Theorem 3.4.5. The variety of regular paralattices is the join of the varieties of skew
lattices and skew* lattices. In particular, every regular paralattice factors as the fibred product
of its maximal skew lattice image with its maximal skew* lattice images over its maximal lattice
image.
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