Page 76 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 76
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Next observe that a∧(c∧b) = a∧(c∧(c∧b)) = (a∧c)∧(c∧b) in A∪C while
(a∧c)∧b = ((a∧c)∧c)∧b = (a∧c)∧(c∧b) in B∪C so that a∧(c∧b) = (a∧c)∧b. Also
c∧(a∧b) = c∧(a∧b)∧b = c∧b in B∪C, while (c∧a)∧b =c∧(c∧a)∧b = c∧b so that
c∧(a∧b) =(c∧a)∧b.
The three other cases of potential associativity under ∧ with a, b and c are left-right
reflections of cases already considered and must also hold. Finally, the six cases involving ∨ are
the duals of the cases considered. Hence, using our assumption about Proj↑(S; ≻), the dual
arguments in all cases for associativity of the ∨-product involving a, b and c in some order will be
successful. Thus (S; , ∨, ∧) is indeed a partial skew lattice.
Conversely, let (S; , ∨, ∧) be a partial skew lattice. Conditional regularity yields
(a∧b∧a)∧c∧(a∧b∧a) = a∧(b∧c∧b)∧a
for all a ≻ b ≻ c in S. Thus, pcpb = pb∧c∧b and Proj↓(S; ≻) is seen to be a category. In similar
fashion, so is Proj↑(S; ≻). £
Combining the above two results we have:
Theorem 2.5.4. A quasi-ordered set (S; ≻) with a covering P of primitive skew lattices is
the primitive covering of a (necessarily unique) skew lattice (S; ∨, ∧) if and only if
i) Both coset projection families, Proj↓(S; ≻) and Proj↑(S; ≻), form categories under the
usual composition of functions.
ii) The equivalence classes of (S; ≻) form a lattice under their usual partial ordering.
iii) Each pair of equivalence classes is orthogonal in both their join and meet classes. £
The case for normal skew lattices
From the perspective of coset bijections and projections, the significant features of
normal skew lattices are (1) that they are strictly categorical and (2) that for each primitive sub-
algebra A > B there is exactly one A-coset in the lower D-class B, namely B, and thus exactly
one projection of A onto B. One thus has a situation like following, where the ϕi are coset
bijections from individual B-cosets Ai in A onto B.
A A1 A2 A3
∨ ≅ ↓≅ ≅ pB = ϕ1 ∪ϕ2 ∪ϕ3
BB
Clearly upward projections are just the upward coset bijections. We thus modify our discussion
of these matters in a way that is more commensurate to the situation for normal skew lattices.
A projective pair is a pair (K, k) where (i) K: A → B is a regular epimorphism of
rectangular skew lattices, that is, a factorization ϕ: J × B ≅ A exists such K o ϕ is the B-coordinate
74
Next observe that a∧(c∧b) = a∧(c∧(c∧b)) = (a∧c)∧(c∧b) in A∪C while
(a∧c)∧b = ((a∧c)∧c)∧b = (a∧c)∧(c∧b) in B∪C so that a∧(c∧b) = (a∧c)∧b. Also
c∧(a∧b) = c∧(a∧b)∧b = c∧b in B∪C, while (c∧a)∧b =c∧(c∧a)∧b = c∧b so that
c∧(a∧b) =(c∧a)∧b.
The three other cases of potential associativity under ∧ with a, b and c are left-right
reflections of cases already considered and must also hold. Finally, the six cases involving ∨ are
the duals of the cases considered. Hence, using our assumption about Proj↑(S; ≻), the dual
arguments in all cases for associativity of the ∨-product involving a, b and c in some order will be
successful. Thus (S; , ∨, ∧) is indeed a partial skew lattice.
Conversely, let (S; , ∨, ∧) be a partial skew lattice. Conditional regularity yields
(a∧b∧a)∧c∧(a∧b∧a) = a∧(b∧c∧b)∧a
for all a ≻ b ≻ c in S. Thus, pcpb = pb∧c∧b and Proj↓(S; ≻) is seen to be a category. In similar
fashion, so is Proj↑(S; ≻). £
Combining the above two results we have:
Theorem 2.5.4. A quasi-ordered set (S; ≻) with a covering P of primitive skew lattices is
the primitive covering of a (necessarily unique) skew lattice (S; ∨, ∧) if and only if
i) Both coset projection families, Proj↓(S; ≻) and Proj↑(S; ≻), form categories under the
usual composition of functions.
ii) The equivalence classes of (S; ≻) form a lattice under their usual partial ordering.
iii) Each pair of equivalence classes is orthogonal in both their join and meet classes. £
The case for normal skew lattices
From the perspective of coset bijections and projections, the significant features of
normal skew lattices are (1) that they are strictly categorical and (2) that for each primitive sub-
algebra A > B there is exactly one A-coset in the lower D-class B, namely B, and thus exactly
one projection of A onto B. One thus has a situation like following, where the ϕi are coset
bijections from individual B-cosets Ai in A onto B.
A A1 A2 A3
∨ ≅ ↓≅ ≅ pB = ϕ1 ∪ϕ2 ∪ϕ3
BB
Clearly upward projections are just the upward coset bijections. We thus modify our discussion
of these matters in a way that is more commensurate to the situation for normal skew lattices.
A projective pair is a pair (K, k) where (i) K: A → B is a regular epimorphism of
rectangular skew lattices, that is, a factorization ϕ: J × B ≅ A exists such K o ϕ is the B-coordinate
74
