Page 75 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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II: Skew Lattices
Returning to primitive skew lattices, let (S; ≻) be a quasi-ordered set. A primitive
covering P of (S; ≻) consists of (1) an assignment of a rectangular skew lattice structure to each
equivalence class A of (S; ≻) and (2) to any comparable pair of equivalence classes A > B a
primitive skew lattice structure is assigned that extends the separate rectangular structures on A
and B in such a way that A > B as separate D-classes. Given a partial skew lattice (S; , ∨, ∧) ,
its canonical primitive covering is the class of all maximal primitive subalgebras of (S; ≻, ∨, ∧).
Given a primitive covering P of a quasi-ordered set (S; ≻), conditional operations ∨ and
∧ are defined on any ≻-comparable pair of elements e and f in S by letting e∨f and e∧f be the join
and meet respectively given in any primitive subalgebra of P containing both. (This subalgebra
is unique if e and f are not equivalent. Otherwise, e and f lie in a common rectangular subalgebra
where ∨ and ∧ are defined.) One can ask: is (S; , ∨, ∧) a partial skew lattice? If “yes”, then P
would be its canonical primitive covering. Put otherwise: when is a primitive covering of given
quasi-ordered set (S; ≻) the canonical primitive covering of some partial skew lattice
(S; , ∨, ∧) on (S; ≻)? Or: when are the induced operations linearly associative?
So let a primitive covering P of (S; ≻) be given and consider equivalence classes A > B.
If Bj is an A-coset in B, then the lower coset projection of A onto Bj is the function pj: A → B
(note that B is the codomain) projecting each element of A onto its unique image in Bj. Clearly
pj(a) = a∧b∧a for any b in Bj. Similarly for each B-coset Ai in A, an upper coset projection
qi: B → A is given by qi(b) = b∨a∨b for any a in Ai. When A = B, set p = q = 1A. We let
Proj↓(S; ≻) [respectively, Proj↑(S; ≻)] denote the family of all lower [upper] coset projections
between comparable equivalence classes of (S; ≻). If composites of lower [upper] coset
projections are also lower [upper] coset projections, then Proj↓(S; ≻) forms the category of lower
coset projections and Proj↑(S; ≻) forms the category of upper coset projections.
Theorem 2.5.3. The partial algebra (S; , ∨, ∧) induced from a primitive covering P
of a quasi-ordered set (S; ≻) is a partial skew lattice precisely when both coset projection
families, Proj↓(S; ≻) and Proj↑(S; ≻), form categories under ordinary composition of functions.
Proof. Suppose that Proj↓(S; ≻) and Proj↑(S; ≻) are categories under the usual composition of
functions. Given comparable classes A > B > C, with a ∈ A, b ∈ B and c ∈ C we first show that
a∧(b∧c) = (a∧b)∧c. First set d = b∧c∧b in C. Using only primitive operations and projections:
a∧(b∧c) = a∧(d∧c) = (a∧d)∧c = pd(a)∧d∧c = pd(a)∧c
and
(a∧b)∧c = (pb[a]∧b)∧c = pc(pb[a]∧b)∧c = pcpb[a]∧ pc[b]∧c == pcpb[a]∧c.
where pb: A → B, pc: B → C and pd: A → C denote coset projections with b, c and d respectively
in their images. By our assumption about Proj↓(S; ≻) the composition pcpb: A → C is either pd or
their images are disjoint in C. Given aʹ ∈ A such that aʹ > b we have pcpb[aʹ] = pc[b] = d. Thus
the images of pcpb and pd overlap and so are equal. Hence a∧(b∧c) = (a∧b)∧c for all a ∈A.
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Returning to primitive skew lattices, let (S; ≻) be a quasi-ordered set. A primitive
covering P of (S; ≻) consists of (1) an assignment of a rectangular skew lattice structure to each
equivalence class A of (S; ≻) and (2) to any comparable pair of equivalence classes A > B a
primitive skew lattice structure is assigned that extends the separate rectangular structures on A
and B in such a way that A > B as separate D-classes. Given a partial skew lattice (S; , ∨, ∧) ,
its canonical primitive covering is the class of all maximal primitive subalgebras of (S; ≻, ∨, ∧).
Given a primitive covering P of a quasi-ordered set (S; ≻), conditional operations ∨ and
∧ are defined on any ≻-comparable pair of elements e and f in S by letting e∨f and e∧f be the join
and meet respectively given in any primitive subalgebra of P containing both. (This subalgebra
is unique if e and f are not equivalent. Otherwise, e and f lie in a common rectangular subalgebra
where ∨ and ∧ are defined.) One can ask: is (S; , ∨, ∧) a partial skew lattice? If “yes”, then P
would be its canonical primitive covering. Put otherwise: when is a primitive covering of given
quasi-ordered set (S; ≻) the canonical primitive covering of some partial skew lattice
(S; , ∨, ∧) on (S; ≻)? Or: when are the induced operations linearly associative?
So let a primitive covering P of (S; ≻) be given and consider equivalence classes A > B.
If Bj is an A-coset in B, then the lower coset projection of A onto Bj is the function pj: A → B
(note that B is the codomain) projecting each element of A onto its unique image in Bj. Clearly
pj(a) = a∧b∧a for any b in Bj. Similarly for each B-coset Ai in A, an upper coset projection
qi: B → A is given by qi(b) = b∨a∨b for any a in Ai. When A = B, set p = q = 1A. We let
Proj↓(S; ≻) [respectively, Proj↑(S; ≻)] denote the family of all lower [upper] coset projections
between comparable equivalence classes of (S; ≻). If composites of lower [upper] coset
projections are also lower [upper] coset projections, then Proj↓(S; ≻) forms the category of lower
coset projections and Proj↑(S; ≻) forms the category of upper coset projections.
Theorem 2.5.3. The partial algebra (S; , ∨, ∧) induced from a primitive covering P
of a quasi-ordered set (S; ≻) is a partial skew lattice precisely when both coset projection
families, Proj↓(S; ≻) and Proj↑(S; ≻), form categories under ordinary composition of functions.
Proof. Suppose that Proj↓(S; ≻) and Proj↑(S; ≻) are categories under the usual composition of
functions. Given comparable classes A > B > C, with a ∈ A, b ∈ B and c ∈ C we first show that
a∧(b∧c) = (a∧b)∧c. First set d = b∧c∧b in C. Using only primitive operations and projections:
a∧(b∧c) = a∧(d∧c) = (a∧d)∧c = pd(a)∧d∧c = pd(a)∧c
and
(a∧b)∧c = (pb[a]∧b)∧c = pc(pb[a]∧b)∧c = pcpb[a]∧ pc[b]∧c == pcpb[a]∧c.
where pb: A → B, pc: B → C and pd: A → C denote coset projections with b, c and d respectively
in their images. By our assumption about Proj↓(S; ≻) the composition pcpb: A → C is either pd or
their images are disjoint in C. Given aʹ ∈ A such that aʹ > b we have pcpb[aʹ] = pc[b] = d. Thus
the images of pcpb and pd overlap and so are equal. Hence a∧(b∧c) = (a∧b)∧c for all a ∈A.
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