Page 61 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 61
II: Skew Lattices
⎧⎡ 0 a1 a2 a3 a4 ... c ⎤ ⎫
⎪⎢ δ1 0 0 0 ⎪
⎪⎢ 0 ... b1 ⎥ δi = 0, 1 ⎪
⎥
⎪⎢ 0 ⎥ ⎪
⎪⎨⎪⎢⎢⎢ 0 δ2 00 ... b2 ⎥ ai = aiδi ⎪
0 0 0 ⎥ bj = b jδ ⎬ .
0 0 δ3 0 ... b3 ⎥ ⎪
⎪⎢ 0 0 δ4 ⎥ j ⎪
⎪⎢ ... b4 ⎥ ⎪
c = ∑ aibi
⎪⎩⎢⎣ 0 0 0 0 0 ... 0 ⎦⎥ ⎪⎭
This is the normal skew lattice E(R) for the upper triangular subring R that it ring-generates. £
Query: Are maximal regular bands in rings skew lattices under ∇? Do regular bands in
rings generate skew lattices under ∇? Do they even generate ∇-bands? That this is not the case
in general is guaranteed by:
Example 2.3.5. In the semigroup ring &[Rega,b,c] of the free regular band on {a, b, c},
the product (a ∇ b)c = ac + bc + bac – abac – babc fails to be idempotent. £
2.4 Primitive skew lattices and skew lattice structure
A primitive skew lattice is a skew lattice P consisting of exactly two D-classes A > B .
Primitive skew lattices and their relation to arbitrary skew lattices were studied in Leech [1993].
We present a number of results from that paper.
Given a primitive skew lattice P with D-classes A > B in P/D, a coset of A in B is any
subset of B of the form A ∧ b ∧ A = {a ∧ b ∧ aʹ⎮a, aʹ ∈A} for some fixed b ∈ B. Similarly, a
coset of B in A is any subset of the form B ∨ a ∨ B = {b ∨ a ∨ bʹ⎮b, bʹ ∈ B} for some a ∈ A.
Since both operations are regular, an alternative description of both types of cosets is given by
A ∧ b ∧ A = {a ∧ b ∧ a⎮a ∈A} and B ∨ a ∨ B = {b ∨ a ∨ b⎮b ∈ B}.
Indeed, {a ∧ b ∧ a⎮a ∈A} ⊆ A ∧ b ∧ A as already defined.. But by regularity,
a ∧ b ∧ aʹ = a ∧ a′ ∧ a ∧ b ∧ a′ ∧ a ∧ a′ = (a ∧ aʹ) ∧ b ∧ (a ∧ aʹ)
which is of the form a ∧ b ∧ a. The case for B ∨ a ∨ B is similar. For any a ∈ A, the set
a ∧ B ∧ a = {a ∧ b ∧ a⎮b ∈ B} = {b ∈ B⎮b ≤ a}
is the image set of a in B. Its elements are the images of a in B. Dually, given any b ∈ B, the set
b ∨ A∨ b = {a ∈ A; a ≥ b} is the image set of b in A. We have the following fundamental result:
59
⎧⎡ 0 a1 a2 a3 a4 ... c ⎤ ⎫
⎪⎢ δ1 0 0 0 ⎪
⎪⎢ 0 ... b1 ⎥ δi = 0, 1 ⎪
⎥
⎪⎢ 0 ⎥ ⎪
⎪⎨⎪⎢⎢⎢ 0 δ2 00 ... b2 ⎥ ai = aiδi ⎪
0 0 0 ⎥ bj = b jδ ⎬ .
0 0 δ3 0 ... b3 ⎥ ⎪
⎪⎢ 0 0 δ4 ⎥ j ⎪
⎪⎢ ... b4 ⎥ ⎪
c = ∑ aibi
⎪⎩⎢⎣ 0 0 0 0 0 ... 0 ⎦⎥ ⎪⎭
This is the normal skew lattice E(R) for the upper triangular subring R that it ring-generates. £
Query: Are maximal regular bands in rings skew lattices under ∇? Do regular bands in
rings generate skew lattices under ∇? Do they even generate ∇-bands? That this is not the case
in general is guaranteed by:
Example 2.3.5. In the semigroup ring &[Rega,b,c] of the free regular band on {a, b, c},
the product (a ∇ b)c = ac + bc + bac – abac – babc fails to be idempotent. £
2.4 Primitive skew lattices and skew lattice structure
A primitive skew lattice is a skew lattice P consisting of exactly two D-classes A > B .
Primitive skew lattices and their relation to arbitrary skew lattices were studied in Leech [1993].
We present a number of results from that paper.
Given a primitive skew lattice P with D-classes A > B in P/D, a coset of A in B is any
subset of B of the form A ∧ b ∧ A = {a ∧ b ∧ aʹ⎮a, aʹ ∈A} for some fixed b ∈ B. Similarly, a
coset of B in A is any subset of the form B ∨ a ∨ B = {b ∨ a ∨ bʹ⎮b, bʹ ∈ B} for some a ∈ A.
Since both operations are regular, an alternative description of both types of cosets is given by
A ∧ b ∧ A = {a ∧ b ∧ a⎮a ∈A} and B ∨ a ∨ B = {b ∨ a ∨ b⎮b ∈ B}.
Indeed, {a ∧ b ∧ a⎮a ∈A} ⊆ A ∧ b ∧ A as already defined.. But by regularity,
a ∧ b ∧ aʹ = a ∧ a′ ∧ a ∧ b ∧ a′ ∧ a ∧ a′ = (a ∧ aʹ) ∧ b ∧ (a ∧ aʹ)
which is of the form a ∧ b ∧ a. The case for B ∨ a ∨ B is similar. For any a ∈ A, the set
a ∧ B ∧ a = {a ∧ b ∧ a⎮b ∈ B} = {b ∈ B⎮b ≤ a}
is the image set of a in B. Its elements are the images of a in B. Dually, given any b ∈ B, the set
b ∨ A∨ b = {a ∈ A; a ≥ b} is the image set of b in A. We have the following fundamental result:
59
