Page 129 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 129
IV: Skew Boolean Algebras
∑1r Di0 is an internal direct product of the primitive subalgebras D10 , … , Dr0 that we call
the orthosum of the Di0 . Of course, ai \ bi = 0 whenever ai D bi. A special case occurs when the
Di are atomic D-classes lying directly over the class {0}. Here ai ∧ bj = 0 for elements ai and bi
from distinct atomic D-classes, making them orthogonal. In general, nonzero orthogonal D-
classes D1, … , Dr are the atomic D-classes of the generated subalgebra ∑1r Di0 . In any case we
have a basic result for skew Boolean algebras with only finitely many D-classes:
Theorem 4.2.2. If S is a nontrivial skew Boolean algebra with finitely many D-classes
and having atomic D-classes {D1, … , Dr}, then S is the orthosum ∑1r Di0 of primitive
subalgebras { D10 , … , Dr0 }. £
The above decomposition is an internal form of the atomic decomposition of a skew
Boolean algebra S, which must occur when S/D is finite. This internal form is unique. The
external form, given as a direct product, is unique to within isomorphism. In the left-handed case
for finite S, the standard atomic decomposition is
S ≅ n1L × n2L × … × nrL with 2 ≤ n1 ≤ n2 ≤ … ≤ nr ,
with nL being the unique left-handed primitive algebra on {0, 1, 2, … , n –1} with 0 being the
0-element. Standard decompositions are also unique. Consider 2 × 2 × 4L × 5L × 5L or more
briefly 22 × 4L × 5L2. In this instance 22 provides the center of the algebra where “L” is
superfluous. Similar remarks hold in the right-handed case. In the two-sided general case one
uses notation such as 3L•5R to represent the primitive algebra 3L×25R given by the fibered
product, as in: S ≅ 23 × 3L•5R × 5L•4R × 7L•7R. In this case a standard decomposition could be
given by lexicographically ordering the factors. In any case, a finite skew Boolean algebra is
classified when its standard atomic decomposition is given.
Example 4.2.1. Partial function algebras serve as primary examples of SBAs. Note that
PL({1, … , n},{1, … , m}) ≅ Πin=1 PL({i}, {1, 2, … , m})
( )≅ P L ({1}, {1, 2, ... , m}) n ≅ (m + 1)nL .
In particular, PL({1, … , n},{1}) ≅ 2n. In this case each partial function f is determined by
choosing a subset of {1, 2, … , n} to be f–1(1), resulting in a bijection between PL({1, … , n},{1})
and the power set of {1, 2, … , n} that preserves the generalized Boolean operations.
We characterize congruences on and homomorphisms between skew Boolean algebras
with finitely many D-classes.
127
∑1r Di0 is an internal direct product of the primitive subalgebras D10 , … , Dr0 that we call
the orthosum of the Di0 . Of course, ai \ bi = 0 whenever ai D bi. A special case occurs when the
Di are atomic D-classes lying directly over the class {0}. Here ai ∧ bj = 0 for elements ai and bi
from distinct atomic D-classes, making them orthogonal. In general, nonzero orthogonal D-
classes D1, … , Dr are the atomic D-classes of the generated subalgebra ∑1r Di0 . In any case we
have a basic result for skew Boolean algebras with only finitely many D-classes:
Theorem 4.2.2. If S is a nontrivial skew Boolean algebra with finitely many D-classes
and having atomic D-classes {D1, … , Dr}, then S is the orthosum ∑1r Di0 of primitive
subalgebras { D10 , … , Dr0 }. £
The above decomposition is an internal form of the atomic decomposition of a skew
Boolean algebra S, which must occur when S/D is finite. This internal form is unique. The
external form, given as a direct product, is unique to within isomorphism. In the left-handed case
for finite S, the standard atomic decomposition is
S ≅ n1L × n2L × … × nrL with 2 ≤ n1 ≤ n2 ≤ … ≤ nr ,
with nL being the unique left-handed primitive algebra on {0, 1, 2, … , n –1} with 0 being the
0-element. Standard decompositions are also unique. Consider 2 × 2 × 4L × 5L × 5L or more
briefly 22 × 4L × 5L2. In this instance 22 provides the center of the algebra where “L” is
superfluous. Similar remarks hold in the right-handed case. In the two-sided general case one
uses notation such as 3L•5R to represent the primitive algebra 3L×25R given by the fibered
product, as in: S ≅ 23 × 3L•5R × 5L•4R × 7L•7R. In this case a standard decomposition could be
given by lexicographically ordering the factors. In any case, a finite skew Boolean algebra is
classified when its standard atomic decomposition is given.
Example 4.2.1. Partial function algebras serve as primary examples of SBAs. Note that
PL({1, … , n},{1, … , m}) ≅ Πin=1 PL({i}, {1, 2, … , m})
( )≅ P L ({1}, {1, 2, ... , m}) n ≅ (m + 1)nL .
In particular, PL({1, … , n},{1}) ≅ 2n. In this case each partial function f is determined by
choosing a subset of {1, 2, … , n} to be f–1(1), resulting in a bijection between PL({1, … , n},{1})
and the power set of {1, 2, … , n} that preserves the generalized Boolean operations.
We characterize congruences on and homomorphisms between skew Boolean algebras
with finitely many D-classes.
127
