Page 86 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 86
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
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higher D-class, but neither nor * separate aʹ from a and so neither is Δ.) If S = A, then S is
rectangular and so, being subdirectly irreducible, is a copy of either R2 or L2. Otherwise lower
cc
D-classes exist and in particular there is an element c < a. Since both and * identify a with
cc
c, both are the trivial congruence, the universal relation ∇. Since and * equal ∇ for all c < a,
A must have a unique singleton lower class, {0} and thus S = A0. Since S is subdirectly
irreducible, A itself is a copy of either R2 or L2 so that S is a copy of either 3R or 3L. £
Corollary 2.6.13. The variety of strongly distributive skew lattices is generated by 5. The
variety of symmetric, normal, skew lattices is generated by 5 plus the variety of lattices. £
We saw in Section 2.3 normal skew lattices arise as maximal normal bands in rings.
Before proceeding to the next section, we offer an example of somewhat different character, an
example that both illustrates much that has occurred in this section and also sets the stage for
developments in Chapter 4.
Example 2.6.1. Given sets A and B, let PR(A, B) be the set of all partial function from
A to B. PR(A, B) becomes a strongly distributive skew lattice upon setting
f ∨ g = f ∪ g⎮G\F and f ∧ g = g⎮F∩G
where F and G denote the actual functional domains in A of f and g respectively. PR(A, B)
factors as ∏a∈A PR({a}, B) with each factor PR({a}, B) being a copy of the primitive algebra B0,
the right rectangular skew lattice on A with 0 adjoined. The various D-classes are indexed by the
subset algebra 2A that forms the maximal lattice image of PR(A, B). For any subset F of A, its
D-class is precisely BF. If G ⊆ F, K(BF , BG ) : BF → BG is just the natural coordinate-wise
projection. If A is finite, P R(A, B) has the primary decomposition ∏ {2A[ Ba Pa ] a ∈ A }
2A
where Pa is the principal filter generated by a. £
Example 2.6.2. Given sets A and B, the left-handed variant of the above skew lattice on
PL(A, B) is given by setting f ∨ g = g ∪ f⎮F\G and f ∧ g = f⎮F∩G. Remarks similar to those
made above can be made here also. £
By a ring of partial functions is meant any subalgebra PR(A, B) for some pair of sets A
and B. The following variation of Theorem 2.6.12 holds. By a full ring of partial functions is
meant a subalgebra S of some PR(A, B) such that each a ∈ A correspond to a subset Ba ⊆ B such
that the primary decomposition of PR(A, B) given above restricts to the primary decomposition
∏ {2A[ Ba Pa ] a ∈ A } .
2A
84
aa
higher D-class, but neither nor * separate aʹ from a and so neither is Δ.) If S = A, then S is
rectangular and so, being subdirectly irreducible, is a copy of either R2 or L2. Otherwise lower
cc
D-classes exist and in particular there is an element c < a. Since both and * identify a with
cc
c, both are the trivial congruence, the universal relation ∇. Since and * equal ∇ for all c < a,
A must have a unique singleton lower class, {0} and thus S = A0. Since S is subdirectly
irreducible, A itself is a copy of either R2 or L2 so that S is a copy of either 3R or 3L. £
Corollary 2.6.13. The variety of strongly distributive skew lattices is generated by 5. The
variety of symmetric, normal, skew lattices is generated by 5 plus the variety of lattices. £
We saw in Section 2.3 normal skew lattices arise as maximal normal bands in rings.
Before proceeding to the next section, we offer an example of somewhat different character, an
example that both illustrates much that has occurred in this section and also sets the stage for
developments in Chapter 4.
Example 2.6.1. Given sets A and B, let PR(A, B) be the set of all partial function from
A to B. PR(A, B) becomes a strongly distributive skew lattice upon setting
f ∨ g = f ∪ g⎮G\F and f ∧ g = g⎮F∩G
where F and G denote the actual functional domains in A of f and g respectively. PR(A, B)
factors as ∏a∈A PR({a}, B) with each factor PR({a}, B) being a copy of the primitive algebra B0,
the right rectangular skew lattice on A with 0 adjoined. The various D-classes are indexed by the
subset algebra 2A that forms the maximal lattice image of PR(A, B). For any subset F of A, its
D-class is precisely BF. If G ⊆ F, K(BF , BG ) : BF → BG is just the natural coordinate-wise
projection. If A is finite, P R(A, B) has the primary decomposition ∏ {2A[ Ba Pa ] a ∈ A }
2A
where Pa is the principal filter generated by a. £
Example 2.6.2. Given sets A and B, the left-handed variant of the above skew lattice on
PL(A, B) is given by setting f ∨ g = g ∪ f⎮F\G and f ∧ g = f⎮F∩G. Remarks similar to those
made above can be made here also. £
By a ring of partial functions is meant any subalgebra PR(A, B) for some pair of sets A
and B. The following variation of Theorem 2.6.12 holds. By a full ring of partial functions is
meant a subalgebra S of some PR(A, B) such that each a ∈ A correspond to a subset Ba ⊆ B such
that the primary decomposition of PR(A, B) given above restricts to the primary decomposition
∏ {2A[ Ba Pa ] a ∈ A } .
2A
84
