Page 229 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 229
VI: Skew Lattices in Rings
Proof. Let D1, D2, … be a complete listing of the D-classes of S. Pick a1 ∈ D1 and consider the
set X = {x ∈S⎮a1x = xa1}. X is a band containing a1 and meeting each D-class of S. Indeed,
given any other D-class Dʹ with y ∈Dʹ we have a1 ≥ a1ya1 ≤ a1ya1sysa1ya1 = b. It follows that
a1b = a1ya1 = ba1 so that b ∈X. Moreover, X is a s-band. Indeed, given x, y ∈ X, then a1 and
xsy = x + y + yx – xyx – yxy clearly commute, so that xsy ∈X also. Note that D1 ∩ X = {a1}.
Next pick a2 ∈ D2 ∩ X, set Y = {y ∈X⎮a2y = ya2}. Again Y is a s-band that meets each D-
class of S and moreover Di ∩ X = {ai} for i = 1, 2. The process continues through the countable
set of D-classes to produce the lattice section T = {a1, a2, a3, …}. By placing the D-classes of
the elements in T0 at the front of the list of D-classes, the second assertion follows. Finally,
suppose that all lattice sections T induce a common associative join ∨ on S. Let e, f ∈ S be given.
Then e and f generate a skew lattice S0 in S with at most 16 elements. (See Theorem 2.7.5 and
the surrounding discussion.) Take a lattice section T0 of S0 and extend it to a full lattice section T
of S, using the first part of this theorem. Then e ∨T f = e ∨T0 f and the latter equals esf in S0
by Theorem 6.2.12. Thus s is the common associative join induced by all lattice sections. £
Thus while “really big” s-bands need not have lattice sections, thanks to Theorem
6.2.13 all s-bands in finite dimensional matrix rings over fields have them so that all of the
theorem applies.
Since s is associative if and only if it is thus on all finitely generated subalgebras of a
given s-band, an alternative to Theorem 6.2.12 is given in the following corollary to the above
theorems and its own following corollary in turn.
Corollary 6.2.14. Given a s-band S, s is associative if and only if for all eDu and fDv
situations in S where uv = vu, esf is calculated as (eu ○ fv)(ue ○ vf) . £
Corollary 6.2.15. s is associative on a s-band S if and only if it is associative on all
sub-algebras generated from at most 2 D-classes. £
The associativity of s: the role of primitive algebras
As with skew lattices, a s-band S is primitive if it has exactly two D-classes, A > B. In
general, a band S is totally pre-ordered if either e ≻ f or f ≻ e for all e, f ∈ S. (Again, e ≻ f if
fef = f, or equivalently for s-bands, esfse = e.) These bands are of interest because maximal,
totally pre-ordered regular bands in a ring form s-bands. (See Corollary 6.3.5 below.) For such
s-bands, the previous results imply that to check for the associativity of s one need only check
for associativity on its primitive subalgebras. It is thus natural to ask if the “totally pre-ordered”
condition can be removed from this observation? Put otherwise, if s is associative on all
primitive subalgebras of a s-band, must it be associative on that band? Or does a s-band S
exist with exactly four D-classes, two of which are mutually incomparable, such that s is
associative on all five maximal primitive subalgebras but not on all of S? This leads us to:
227
Proof. Let D1, D2, … be a complete listing of the D-classes of S. Pick a1 ∈ D1 and consider the
set X = {x ∈S⎮a1x = xa1}. X is a band containing a1 and meeting each D-class of S. Indeed,
given any other D-class Dʹ with y ∈Dʹ we have a1 ≥ a1ya1 ≤ a1ya1sysa1ya1 = b. It follows that
a1b = a1ya1 = ba1 so that b ∈X. Moreover, X is a s-band. Indeed, given x, y ∈ X, then a1 and
xsy = x + y + yx – xyx – yxy clearly commute, so that xsy ∈X also. Note that D1 ∩ X = {a1}.
Next pick a2 ∈ D2 ∩ X, set Y = {y ∈X⎮a2y = ya2}. Again Y is a s-band that meets each D-
class of S and moreover Di ∩ X = {ai} for i = 1, 2. The process continues through the countable
set of D-classes to produce the lattice section T = {a1, a2, a3, …}. By placing the D-classes of
the elements in T0 at the front of the list of D-classes, the second assertion follows. Finally,
suppose that all lattice sections T induce a common associative join ∨ on S. Let e, f ∈ S be given.
Then e and f generate a skew lattice S0 in S with at most 16 elements. (See Theorem 2.7.5 and
the surrounding discussion.) Take a lattice section T0 of S0 and extend it to a full lattice section T
of S, using the first part of this theorem. Then e ∨T f = e ∨T0 f and the latter equals esf in S0
by Theorem 6.2.12. Thus s is the common associative join induced by all lattice sections. £
Thus while “really big” s-bands need not have lattice sections, thanks to Theorem
6.2.13 all s-bands in finite dimensional matrix rings over fields have them so that all of the
theorem applies.
Since s is associative if and only if it is thus on all finitely generated subalgebras of a
given s-band, an alternative to Theorem 6.2.12 is given in the following corollary to the above
theorems and its own following corollary in turn.
Corollary 6.2.14. Given a s-band S, s is associative if and only if for all eDu and fDv
situations in S where uv = vu, esf is calculated as (eu ○ fv)(ue ○ vf) . £
Corollary 6.2.15. s is associative on a s-band S if and only if it is associative on all
sub-algebras generated from at most 2 D-classes. £
The associativity of s: the role of primitive algebras
As with skew lattices, a s-band S is primitive if it has exactly two D-classes, A > B. In
general, a band S is totally pre-ordered if either e ≻ f or f ≻ e for all e, f ∈ S. (Again, e ≻ f if
fef = f, or equivalently for s-bands, esfse = e.) These bands are of interest because maximal,
totally pre-ordered regular bands in a ring form s-bands. (See Corollary 6.3.5 below.) For such
s-bands, the previous results imply that to check for the associativity of s one need only check
for associativity on its primitive subalgebras. It is thus natural to ask if the “totally pre-ordered”
condition can be removed from this observation? Put otherwise, if s is associative on all
primitive subalgebras of a s-band, must it be associative on that band? Or does a s-band S
exist with exactly four D-classes, two of which are mutually incomparable, such that s is
associative on all five maximal primitive subalgebras but not on all of S? This leads us to:
227
