Page 26 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 26
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
and only if it satisfies yzx = zyx, also making it normal. Since any regular band S can be
embedded in S/R × S/L, with the latter being normal if and only if S is, the corollary follows.
The remainder of this section is devoted to assorted further remarks about bands. We
begin by defining four canonical sub-bands arising for each e in a band S.
e↓ ={f ∈ S ⎢f ≤ e}, e⇓ = {f ∈ S ⎢f ≺ e}, L⇓e = {f ∈ S ⎢f L≺ e} and e⇓R = {f ∈ S ⎢f R≺ e}.
Lemma 1.2.10. Given a normal band S, for each e ∈S, under the given operation e↓ is a
semilattice; conversely, every band satisfying this property is normal.
Proof. If S is normal and f, g ≤ e in S, then fg = efge = egfe = gf. Thus e↓ is a commutative sub-
band of S, that is, a semilattice in S. Conversely, if each e↓ is commutative, then S at least
satisfies the identity xyxzx = xzxyx. From this we derive the identity xyzyxzyzx = xzyzxyzyx. But
since xyzy, zyzx and yxz lie in the same D-class, xyzyxzyzx reduces to xyzyzx = xyzx. Similarly,
xzyzxyzyx reduces to xzyx and xyzx = xzyx follows. £
Given D-classes A, B in a band S we write A ≥ B if a ≻ b for any (and hence all) pairs a
∈ A and b ∈ B. When A ≥ B but A ≠ B, we write A > B. This reflects, of course, what occurs
between the corresponding elements in the underlying lattice S/D. When A ≥ B we say that A
and B are comparable D-classes. The following results provide an explicit description of the
architecture of a normal band.
Lemma 1.2.11. Given D-classes A ≥ B in a normal band S, for each a ∈A exactly one b
∈B exists such that a ≥ b. The function α: A → B determined by α(a) = b if a ≥ b is a
homomorphism of rectangular bands; moreover α(a) = aba for all b ∈ B. Conversely, if ≥
induces functions in this manner between all pairs of comparable D-classes of a band S, then S is
normal.
Proof. Given a ∈ A and b ∈ B for comparable D-classes A ≥ B in a normal band S, observe that
a ≥ aba in B. Given bʹ ∈ B also, then bbʹb = b and bʹbbʹ = bʹ in B. Normality give us
aba = abbʹba = abʹba = abʹbʹba = abʹbbʹa = abʹbʹa = abʹa.
Thus the procedure a → aba induces a well-defined map from A to B such that a ≥ aba with aba
being independent of b ∈ B. Since b = aba whenever a ≥ b, the first statement of the lemma
follows. That α is a homomorphism follows immediately from S being normal: if a, aʹ ∈ A and
b ∈ B, then abaaʹbaʹ = aaʹbbaaʹ aaʹbaaʹ. Conversely, suppose that ≥ always induces a function
between comparable pairs of D-classes of a band S in the above manner. Since xyzx D xzyx with
x ≥ xyzx and x ≥ xzyx for all x, y, z ∈ S, this assumption gives xyzx = xzyx in S. S is thus normal
by Corollary 1.2.9. £
24
and only if it satisfies yzx = zyx, also making it normal. Since any regular band S can be
embedded in S/R × S/L, with the latter being normal if and only if S is, the corollary follows.
The remainder of this section is devoted to assorted further remarks about bands. We
begin by defining four canonical sub-bands arising for each e in a band S.
e↓ ={f ∈ S ⎢f ≤ e}, e⇓ = {f ∈ S ⎢f ≺ e}, L⇓e = {f ∈ S ⎢f L≺ e} and e⇓R = {f ∈ S ⎢f R≺ e}.
Lemma 1.2.10. Given a normal band S, for each e ∈S, under the given operation e↓ is a
semilattice; conversely, every band satisfying this property is normal.
Proof. If S is normal and f, g ≤ e in S, then fg = efge = egfe = gf. Thus e↓ is a commutative sub-
band of S, that is, a semilattice in S. Conversely, if each e↓ is commutative, then S at least
satisfies the identity xyxzx = xzxyx. From this we derive the identity xyzyxzyzx = xzyzxyzyx. But
since xyzy, zyzx and yxz lie in the same D-class, xyzyxzyzx reduces to xyzyzx = xyzx. Similarly,
xzyzxyzyx reduces to xzyx and xyzx = xzyx follows. £
Given D-classes A, B in a band S we write A ≥ B if a ≻ b for any (and hence all) pairs a
∈ A and b ∈ B. When A ≥ B but A ≠ B, we write A > B. This reflects, of course, what occurs
between the corresponding elements in the underlying lattice S/D. When A ≥ B we say that A
and B are comparable D-classes. The following results provide an explicit description of the
architecture of a normal band.
Lemma 1.2.11. Given D-classes A ≥ B in a normal band S, for each a ∈A exactly one b
∈B exists such that a ≥ b. The function α: A → B determined by α(a) = b if a ≥ b is a
homomorphism of rectangular bands; moreover α(a) = aba for all b ∈ B. Conversely, if ≥
induces functions in this manner between all pairs of comparable D-classes of a band S, then S is
normal.
Proof. Given a ∈ A and b ∈ B for comparable D-classes A ≥ B in a normal band S, observe that
a ≥ aba in B. Given bʹ ∈ B also, then bbʹb = b and bʹbbʹ = bʹ in B. Normality give us
aba = abbʹba = abʹba = abʹbʹba = abʹbbʹa = abʹbʹa = abʹa.
Thus the procedure a → aba induces a well-defined map from A to B such that a ≥ aba with aba
being independent of b ∈ B. Since b = aba whenever a ≥ b, the first statement of the lemma
follows. That α is a homomorphism follows immediately from S being normal: if a, aʹ ∈ A and
b ∈ B, then abaaʹbaʹ = aaʹbbaaʹ aaʹbaaʹ. Conversely, suppose that ≥ always induces a function
between comparable pairs of D-classes of a band S in the above manner. Since xyzx D xzyx with
x ≥ xyzx and x ≥ xzyx for all x, y, z ∈ S, this assumption gives xyzx = xzyx in S. S is thus normal
by Corollary 1.2.9. £
24
