Page 137 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 137
IV: Skew Boolean Algebras
Theorem 4.3.1. A skew lattice can be embedded in a skew Boolean algebra if and only if
it is strongly distributive.
Proof. (Only if) Skew lattice reducts of skew Boolean algebras are strongly distributive and
strongly distributive skew lattices form a skew lattice subvariety. (If) By Theorem 2.6.12 every
strongly distributive skew lattice can be embedded in a power of the skew lattice 5, which
however, is the reduct of that power of the skew Boolean algebra 5. £
Theorem 4.3.2. For skew lattice S the following are equivalent:
i) S can be embedded in a right-handed skew Boolean algebra.
ii) S is right-handed and strongly distributive.
iii) S can be embedded in a slew lattice of partial functions PR(A, B).
Proof. The equivalence of (i) and (ii) is seen as in the previous proof, but with 5 replaced by 3.
If S can be embedded in some partial function skew lattice PR(A, B), then since the latter is right-
handed and strongly distributive so it S. The converse follows from Theorem 2.6.14. £
Theorem 4.3.3. Given an identity in ∨ and ∧ the following are equivalent:
i) The identity holds in all skew Boolean algebras.
ii) The identity holds in all strongly distributive skew lattices.
iii) The identity holds in the skew lattice 5.
Proof. Both (i) and (ii) are equivalent by Theorem 4.2.1, and each clearly implies (iii).
Conversely, any identity holding on 5 must also hold on any sub-skew lattice of any power of 5,
and extending via isomorphism, it must hold on any strongly distributive skew lattice. £
Of course, one has the right-handed specialization of the above.
Theorem 4.3.3R. Given an identity in ∨ and ∧ the following are equivalent:
i) The identity holds in all right-handed skew Boolean algebras.
ii) The identity holds all right-handed strongly distributive skew lattices.
iii) The identity holds in the skew lattice 3R.
iv) The identity holds in all partial function algebras P(A, B). £
Proof. The equivalence of (i) – (iii) is seen similarly as in the general case. Their equivalence
with (iv) comes from Theorem 2.6.14. £
135
Theorem 4.3.1. A skew lattice can be embedded in a skew Boolean algebra if and only if
it is strongly distributive.
Proof. (Only if) Skew lattice reducts of skew Boolean algebras are strongly distributive and
strongly distributive skew lattices form a skew lattice subvariety. (If) By Theorem 2.6.12 every
strongly distributive skew lattice can be embedded in a power of the skew lattice 5, which
however, is the reduct of that power of the skew Boolean algebra 5. £
Theorem 4.3.2. For skew lattice S the following are equivalent:
i) S can be embedded in a right-handed skew Boolean algebra.
ii) S is right-handed and strongly distributive.
iii) S can be embedded in a slew lattice of partial functions PR(A, B).
Proof. The equivalence of (i) and (ii) is seen as in the previous proof, but with 5 replaced by 3.
If S can be embedded in some partial function skew lattice PR(A, B), then since the latter is right-
handed and strongly distributive so it S. The converse follows from Theorem 2.6.14. £
Theorem 4.3.3. Given an identity in ∨ and ∧ the following are equivalent:
i) The identity holds in all skew Boolean algebras.
ii) The identity holds in all strongly distributive skew lattices.
iii) The identity holds in the skew lattice 5.
Proof. Both (i) and (ii) are equivalent by Theorem 4.2.1, and each clearly implies (iii).
Conversely, any identity holding on 5 must also hold on any sub-skew lattice of any power of 5,
and extending via isomorphism, it must hold on any strongly distributive skew lattice. £
Of course, one has the right-handed specialization of the above.
Theorem 4.3.3R. Given an identity in ∨ and ∧ the following are equivalent:
i) The identity holds in all right-handed skew Boolean algebras.
ii) The identity holds all right-handed strongly distributive skew lattices.
iii) The identity holds in the skew lattice 3R.
iv) The identity holds in all partial function algebras P(A, B). £
Proof. The equivalence of (i) – (iii) is seen similarly as in the general case. Their equivalence
with (iv) comes from Theorem 2.6.14. £
135
