Page 265 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 265
VII: Further Topics in Skew Boolean Algebras
x∩x = x/(x/x) = x/0 = x; x∩0 = x/(x/0) = x/x = 0; and x∩y = y∩x by (g).
With a bit more work one sees that (S; ∩, 0) is a meet semilattice with a minimum 0 such that
each principal ideal ⎡x⎤ = {y⎪y ≤ x} forms a Boolean lattice, with x / y being the complement of
x∩y in ⎡x⎤. For any pair u, v ∈ ⎡x⎤, u ∨ v is given as x / [(x/u) ∩ (x/v)]. (See the references above.)
The algebra (S; ∩, /, 0) is sometimes called a Boolean semilattice.
This is relevant to skew Boolean ∩-algebras. Given such an algebra (S: ∨, ∧, \, ∩, 0),
define the iBCK difference e / f on S by
e / f = e \ e∩f.
For all e and f both differences agree, that is e / f = e \ f, if and only if e and f commute. Indeed
one has e \ e∧f∧e = e \ e∩f if and only if e∧f∧e = e∩f. But then e∧f = e∧f∧e∧f = (e∩f) ∧ f = e∩f
and likewise, f∧e = e∩f. The converse is clear. In particular, e / f = e \ f, if e ≥ f. Both ∩ and the
skew Boolean difference \ can be recovered from the iBCK difference / by
e∩f = e / (e / f) and e \ f = e / (e∧f∧e).
Thus skew Boolean ∩-algebras can be viewed as algebras with three binary operations: ∨, ∧, and
/, plus a constant, 0. What identities involving {∨, ∧, /, 0} characterize such algebras? Our
remarks above, together with an examination of what occurs in the primitive case, yield a Second
Signature Bisection Theorem due to Bignall and Leech [1995].
Theorem 7.1.2. Every skew Boolean ∩-algebra (S: ∨, ∧, \, ∩, 0) is term equivalent to an
algebra, (S; ∨, ∧, /, 0) of type 〈2, 2, 2, 0〉 where:
i) (S; ∨, ∧, 0) is a symmetric, normal skew lattice with 0.
ii) (S; /, 0) is an iBCK-algebra.
iii) The induced meet, e∩f = e / (e / f), of (S; /, 0) satisfies the identities of Lemma 4.3.2:
e ∩ (e∧f∧e) = e∧f∧e and e ∧ (e∩f) = e∩f = (e∩f) ∧ e.
Proof. Given a skew Boolean ∩-algebra, (i) is clear. The reduct (S; /, 0), where e / f = e \ e∩f,
satisfies the conditions for an iBCK algebra on primitive skew Boolean algebras and hence on all
skew Boolean ∩-algebras so that (ii) follows. Finally, the iBCK meet e / (e / f) reduces to the
natural intersection ∩. Indeed since both e/f, e∩f ≤ e, we get
e / (e / f) = e \ (e / f) = e \ (e \ e∩f) = e∩f,
and (iii) follows. Conversely, given (i), (S; ∨, ∧, 0) is a symmetric, normal skew lattice with zero
0. By (iii) it shares a common natural partial order ≤ with the /-induced meet ∩. This forces both
algebras to share a common natural meet ∩. Each common principal ideal ⎡e⎤ = {y⎪y ≤ e} is a
Boolean lattice by (ii), which forces (S/D; ∨, ∧) to be a generalized Boolean lattice. It follows
that (S; ∨, ∧, ∩, 0) is an implicit skew Boolean ∩-algebra with iBCK difference /. £
263
x∩x = x/(x/x) = x/0 = x; x∩0 = x/(x/0) = x/x = 0; and x∩y = y∩x by (g).
With a bit more work one sees that (S; ∩, 0) is a meet semilattice with a minimum 0 such that
each principal ideal ⎡x⎤ = {y⎪y ≤ x} forms a Boolean lattice, with x / y being the complement of
x∩y in ⎡x⎤. For any pair u, v ∈ ⎡x⎤, u ∨ v is given as x / [(x/u) ∩ (x/v)]. (See the references above.)
The algebra (S; ∩, /, 0) is sometimes called a Boolean semilattice.
This is relevant to skew Boolean ∩-algebras. Given such an algebra (S: ∨, ∧, \, ∩, 0),
define the iBCK difference e / f on S by
e / f = e \ e∩f.
For all e and f both differences agree, that is e / f = e \ f, if and only if e and f commute. Indeed
one has e \ e∧f∧e = e \ e∩f if and only if e∧f∧e = e∩f. But then e∧f = e∧f∧e∧f = (e∩f) ∧ f = e∩f
and likewise, f∧e = e∩f. The converse is clear. In particular, e / f = e \ f, if e ≥ f. Both ∩ and the
skew Boolean difference \ can be recovered from the iBCK difference / by
e∩f = e / (e / f) and e \ f = e / (e∧f∧e).
Thus skew Boolean ∩-algebras can be viewed as algebras with three binary operations: ∨, ∧, and
/, plus a constant, 0. What identities involving {∨, ∧, /, 0} characterize such algebras? Our
remarks above, together with an examination of what occurs in the primitive case, yield a Second
Signature Bisection Theorem due to Bignall and Leech [1995].
Theorem 7.1.2. Every skew Boolean ∩-algebra (S: ∨, ∧, \, ∩, 0) is term equivalent to an
algebra, (S; ∨, ∧, /, 0) of type 〈2, 2, 2, 0〉 where:
i) (S; ∨, ∧, 0) is a symmetric, normal skew lattice with 0.
ii) (S; /, 0) is an iBCK-algebra.
iii) The induced meet, e∩f = e / (e / f), of (S; /, 0) satisfies the identities of Lemma 4.3.2:
e ∩ (e∧f∧e) = e∧f∧e and e ∧ (e∩f) = e∩f = (e∩f) ∧ e.
Proof. Given a skew Boolean ∩-algebra, (i) is clear. The reduct (S; /, 0), where e / f = e \ e∩f,
satisfies the conditions for an iBCK algebra on primitive skew Boolean algebras and hence on all
skew Boolean ∩-algebras so that (ii) follows. Finally, the iBCK meet e / (e / f) reduces to the
natural intersection ∩. Indeed since both e/f, e∩f ≤ e, we get
e / (e / f) = e \ (e / f) = e \ (e \ e∩f) = e∩f,
and (iii) follows. Conversely, given (i), (S; ∨, ∧, 0) is a symmetric, normal skew lattice with zero
0. By (iii) it shares a common natural partial order ≤ with the /-induced meet ∩. This forces both
algebras to share a common natural meet ∩. Each common principal ideal ⎡e⎤ = {y⎪y ≤ e} is a
Boolean lattice by (ii), which forces (S/D; ∨, ∧) to be a generalized Boolean lattice. It follows
that (S; ∨, ∧, ∩, 0) is an implicit skew Boolean ∩-algebra with iBCK difference /. £
263
