Page 62 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 62
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Theorem 2.4.1. Let P be a primitive skew lattice with D-classes A > B . Then
(1) B is partitioned by the cosets of A in B. In particular, b ∈ A∧b∧A for all b ∈ B
and if x ∈ A∧b∧A for some x ∈ B, then A∧x∧A = A∧b∧A.
(2) The image set in B of any a ∈ A is a transversal of the cosets of A in B.
(3) Dual remarks hold for cosets and element images of B in A. Furthermore:
(4) Given cosets B∨a∨B in A and A∧b∧A in B a natural bijection of both cosets is
given by the natural partial ordering: x in B∨a∨B corresponds to y in A∧b∧A if
and only if x ≥ y.
(5) The meet and join on P are determined jointly by these coset bijections and the
rectangular structure of each D-class.
Proof. By absorption, b = (a∨b)∧b∧(b∨a) for all a in A so that b ∈ A∧b∧A. Given x ∈ A∧b∧A,
say x = m∧b∧n for m, n ∈ A, then for all a ∈ A, a ∧ x ∧ a = a ∧ m ∧ b ∧ n ∧ a = a ∧ b ∧ a where
the second identity holds since ∧ is regular and m, n ≻ a, b. Thus (1) is seen and this conditional
identity also gives us (2). Condition (3) follows by duality. Given cosets B∨a∨B and A∧b∧A,
for any x in B∨a∨B, by (2) x ∧ b ∧ x is the unique element y of A∧b∧A such that x ≥ y. Dually,
for each y in A∧b∧A, y∨a∨y is the unique element x in B∨a∨B such that x ≥ y. Between these
two cosets, the processes x → y ≤ x and y → x ≥ y are reciprocal and (4) is seen. Finally, given
x, y ∈ P, whenever x D y then both x∧y and x∨y are given by the rectangular structure of the
common D-class of x and y. Otherwise, say x ∈ A and y ∈ B, we have x∨y = x∨(y∨x∨y),
y∨x = (y∨x∨y)∨x, x∧y = (x∧y∧x)∧y and y∧x = y∧(x∧y∧x). Since y∨x∨y is the image of y in the
B-coset in A containing x and x∧y∧x is the image of x in the A-coset in B containing y, (5) is
seen. £
Given A > B as above, if A is partitioned by cosets {Ai⎮i ∈ I} and B is partitioned by
cosets {Bj⎮j ∈ J}, then for each pair of indices i, j let ϕji: Ai → Bj denote the coset bijection
given by setting φji(x) = y if for x in Ai, y is the unique element in Bj such that x ≥ y. Then for all
x ∈ Ai and y ∈ Bj
x ∨ y = x ∨ ϕji–1(y), y ∨ x = ϕji–1(y) ∨ x, x ∧ y = ϕji(x) ∧ y and y ∧ x = y ∧ ϕji(x).
Thus it seems that any primitive skew lattice should be obtained by a fairly simple construction.
To this end we begin by calling a right-handed primitive skew lattice right primitive, with left
primitive skew lattices defined in dual fashion. For right primitive skew lattices the description
of a coset can be simplified as
A ∧ b ∧ A = b ∧ A = {b ∧ a⎮a ∈ A} and B ∨ a ∨ B = B ∨ a = {b ∨ a⎮b ∈ B}
for any a ∈ A and b ∈ B since a∧b∧a = b∧a and b∨a∨b = b∨a. Dual remarks hold in the left-
primitive case. All right [left] primitive skew lattices arise as follows.
A P-graph is a pair of partitioned disjoint sets A = ∪i Ai∈ Ι and B = ∪j Bj∈ ϑ , where all Ai
and all Bj have a common cardinality, together with a fixed set of bijections φji: Ai → Bj . The
P-graph is denoted by the triple (Ai, φji, Bj). A right primitive structure is induced on A∪B in the
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Theorem 2.4.1. Let P be a primitive skew lattice with D-classes A > B . Then
(1) B is partitioned by the cosets of A in B. In particular, b ∈ A∧b∧A for all b ∈ B
and if x ∈ A∧b∧A for some x ∈ B, then A∧x∧A = A∧b∧A.
(2) The image set in B of any a ∈ A is a transversal of the cosets of A in B.
(3) Dual remarks hold for cosets and element images of B in A. Furthermore:
(4) Given cosets B∨a∨B in A and A∧b∧A in B a natural bijection of both cosets is
given by the natural partial ordering: x in B∨a∨B corresponds to y in A∧b∧A if
and only if x ≥ y.
(5) The meet and join on P are determined jointly by these coset bijections and the
rectangular structure of each D-class.
Proof. By absorption, b = (a∨b)∧b∧(b∨a) for all a in A so that b ∈ A∧b∧A. Given x ∈ A∧b∧A,
say x = m∧b∧n for m, n ∈ A, then for all a ∈ A, a ∧ x ∧ a = a ∧ m ∧ b ∧ n ∧ a = a ∧ b ∧ a where
the second identity holds since ∧ is regular and m, n ≻ a, b. Thus (1) is seen and this conditional
identity also gives us (2). Condition (3) follows by duality. Given cosets B∨a∨B and A∧b∧A,
for any x in B∨a∨B, by (2) x ∧ b ∧ x is the unique element y of A∧b∧A such that x ≥ y. Dually,
for each y in A∧b∧A, y∨a∨y is the unique element x in B∨a∨B such that x ≥ y. Between these
two cosets, the processes x → y ≤ x and y → x ≥ y are reciprocal and (4) is seen. Finally, given
x, y ∈ P, whenever x D y then both x∧y and x∨y are given by the rectangular structure of the
common D-class of x and y. Otherwise, say x ∈ A and y ∈ B, we have x∨y = x∨(y∨x∨y),
y∨x = (y∨x∨y)∨x, x∧y = (x∧y∧x)∧y and y∧x = y∧(x∧y∧x). Since y∨x∨y is the image of y in the
B-coset in A containing x and x∧y∧x is the image of x in the A-coset in B containing y, (5) is
seen. £
Given A > B as above, if A is partitioned by cosets {Ai⎮i ∈ I} and B is partitioned by
cosets {Bj⎮j ∈ J}, then for each pair of indices i, j let ϕji: Ai → Bj denote the coset bijection
given by setting φji(x) = y if for x in Ai, y is the unique element in Bj such that x ≥ y. Then for all
x ∈ Ai and y ∈ Bj
x ∨ y = x ∨ ϕji–1(y), y ∨ x = ϕji–1(y) ∨ x, x ∧ y = ϕji(x) ∧ y and y ∧ x = y ∧ ϕji(x).
Thus it seems that any primitive skew lattice should be obtained by a fairly simple construction.
To this end we begin by calling a right-handed primitive skew lattice right primitive, with left
primitive skew lattices defined in dual fashion. For right primitive skew lattices the description
of a coset can be simplified as
A ∧ b ∧ A = b ∧ A = {b ∧ a⎮a ∈ A} and B ∨ a ∨ B = B ∨ a = {b ∨ a⎮b ∈ B}
for any a ∈ A and b ∈ B since a∧b∧a = b∧a and b∨a∨b = b∨a. Dual remarks hold in the left-
primitive case. All right [left] primitive skew lattices arise as follows.
A P-graph is a pair of partitioned disjoint sets A = ∪i Ai∈ Ι and B = ∪j Bj∈ ϑ , where all Ai
and all Bj have a common cardinality, together with a fixed set of bijections φji: Ai → Bj . The
P-graph is denoted by the triple (Ai, φji, Bj). A right primitive structure is induced on A∪B in the
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