Page 134 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 134
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
We need to show that this subalgebra is in fact all of LSBAn. We do so by showing that each
generator xk of LSBAn is in the generated subalgebra. The identities above give us:
x1 = (x1∧x2) + (x1\ x2) = (x1∧x2∧x3) + ((x1∧x2)\x3) + ((x1\x2)∧x3) + (x1\x2)\x3)
= (x1∧x2∧x3) + ((x1∧x2)\x3) + ((x1∧x3)\x2) + (x1\(x2∨x3))
= (x1∧x2∧x3∧x4) + ((x1∧x2∧x3)\x4) + …
The process keeps repeating on each new term until generator x1 is resolved into an orthosum of
2n–1 {L|M}-type terms – indeed into all the {L|M}-type terms with left-most entry x1. Similar
calculations work for the remaining generators. Thus the 2n–1 distinct {L|M}-classes are all the
atomic D-classes of LSBAn. £
In the generalized Boolean case, all atomic terms resulting from the same {L|M}
decomposition are equated. Thus the particular left-most generator/variable no longer
differentiates among outcomes. In the 2-sided case, in the Kimura fibered product construction
each left-handed atomic class is matched off with the right-handed atomic class with the same
{L|M} partition. In this case the data of ((y1∧…∧yk), (yʹ1∧…∧yʹk)) can be combined as
y1∧…∧yk∧yʹ1∧…∧yʹk and then reduced via 2-sided normality. Returning to the previous example:
Example 4.2.2 continued. These previous terms describe the atomic classes of both the
left- and right-handed free algebras on {x, y, z, w}. E.g. {(x∧y)\(z∨w), (y∧x)\(z∨w)} works in the
left-hand case, while {(x∧y)\(z∨w), (y∧x)\(z∨w)} works in the right-hand case. In both finite
cases it is possible to describe the “atomic” terms using cyclic permutations in a way that the
terms do double duty. But that won’t “stretch” to the 2-sided case. Here we adjoin both
(x∧y∧x)\(z∨w) and (y∧x∧y)\(z∨w) to the class to get:
{(x∧y)\(z∨w), (y∧x)\(z∨w), (x∧y∧x)\(z∨w), (y∧x∧y)\(z∨w)}.
For two terms to be equal in value, both end variables in the left part would have to agree. In
general, the corresponding atomic classes would be squared in size. £
We thus obtain precise structural descriptions of all four relevant free algebras. In what
follows D{L|M} is the {L|M}-induced D = L-class, PL{L|M} is the left-handed primitive algebra
D{L|M}0 and PR{L|M} is its right-handed counterpart. Also, given primitive algebras P and Q, P•Q
denotes their fibered product over 2, P×2Q. In the next theorem, the trivial algebra 1 on {0} is
included to allow the full distribution of binomial coefficients. This factor corresponds to the
front-empty partition {∅|X}. Hence:
132
We need to show that this subalgebra is in fact all of LSBAn. We do so by showing that each
generator xk of LSBAn is in the generated subalgebra. The identities above give us:
x1 = (x1∧x2) + (x1\ x2) = (x1∧x2∧x3) + ((x1∧x2)\x3) + ((x1\x2)∧x3) + (x1\x2)\x3)
= (x1∧x2∧x3) + ((x1∧x2)\x3) + ((x1∧x3)\x2) + (x1\(x2∨x3))
= (x1∧x2∧x3∧x4) + ((x1∧x2∧x3)\x4) + …
The process keeps repeating on each new term until generator x1 is resolved into an orthosum of
2n–1 {L|M}-type terms – indeed into all the {L|M}-type terms with left-most entry x1. Similar
calculations work for the remaining generators. Thus the 2n–1 distinct {L|M}-classes are all the
atomic D-classes of LSBAn. £
In the generalized Boolean case, all atomic terms resulting from the same {L|M}
decomposition are equated. Thus the particular left-most generator/variable no longer
differentiates among outcomes. In the 2-sided case, in the Kimura fibered product construction
each left-handed atomic class is matched off with the right-handed atomic class with the same
{L|M} partition. In this case the data of ((y1∧…∧yk), (yʹ1∧…∧yʹk)) can be combined as
y1∧…∧yk∧yʹ1∧…∧yʹk and then reduced via 2-sided normality. Returning to the previous example:
Example 4.2.2 continued. These previous terms describe the atomic classes of both the
left- and right-handed free algebras on {x, y, z, w}. E.g. {(x∧y)\(z∨w), (y∧x)\(z∨w)} works in the
left-hand case, while {(x∧y)\(z∨w), (y∧x)\(z∨w)} works in the right-hand case. In both finite
cases it is possible to describe the “atomic” terms using cyclic permutations in a way that the
terms do double duty. But that won’t “stretch” to the 2-sided case. Here we adjoin both
(x∧y∧x)\(z∨w) and (y∧x∧y)\(z∨w) to the class to get:
{(x∧y)\(z∨w), (y∧x)\(z∨w), (x∧y∧x)\(z∨w), (y∧x∧y)\(z∨w)}.
For two terms to be equal in value, both end variables in the left part would have to agree. In
general, the corresponding atomic classes would be squared in size. £
We thus obtain precise structural descriptions of all four relevant free algebras. In what
follows D{L|M} is the {L|M}-induced D = L-class, PL{L|M} is the left-handed primitive algebra
D{L|M}0 and PR{L|M} is its right-handed counterpart. Also, given primitive algebras P and Q, P•Q
denotes their fibered product over 2, P×2Q. In the next theorem, the trivial algebra 1 on {0} is
included to allow the full distribution of binomial coefficients. This factor corresponds to the
front-empty partition {∅|X}. Hence:
132
