Page 147 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 147
IV: Skew Boolean Algebras
xʺ, yʺ ∈S/R. Since S/L and S/R have finite intersections, xʹ∩yʹ and xʺ∩yʺ exist in S/L and S/R,
respectively. If xʹ∩yʹ and xʺ∩yʺ share a common image in S/D, then (xʹ, xʺ) ∩ (yʹ, yʺ) is just
(xʹ∩yʹ, xʺ∩yʺ). In general, let u0∧v0 be the meet in S/D of the respective images u0 of xʹ∩yʹ and
v0 of xʺ∩yʺ in S/D. In the respective D classes of S/L and S/R indexed by u0∧v0, unique
elements wʹ and wʺ exist (by normality) such that both xʹ∩yʹ ≥ wʹ in S/L and xʺ∩yʺ ≥ wʺ in S/R.
The intersection (xʹ, xʺ) ∩ (yʹ, yʺ) is then precisely (wʹ, wʺ). £
The practical consequence of all this as follows: just as all skew Boolean algebras can be
constructed in principle from pairs of left- and right-handed skew Boolean algebras (SL, SR)
having a common maximal commutative Boolean image B using the fibered product, SL ×B SR, so
also all skew Boolean ∩-algebras can be constructed from pairs of left- and right-handed skew
Boolean ∩-algebras (SL, SR) with a common maximal generalized Boolean algebra image B
(with SL and SR viewed as skew Boolean algebras) by exactly the same process, SL ×B SR, even
when their corresponding ∩-outcomes may have distinct locations relative to B. It follows that
the study of skew Boolean ∩-algebras can, in principle, be reduced to studying right-handed skew
Boolean ∩-algebras, or their term-equivalent left-handed duals, since pairs sharing common
maximal lattice images can be spliced together at will. All this is illustrated in the following case
of infinite free skew Boolean algebras.
Atom splitting and the case of infinite free skew Boolean algebras
Consider the inclusion LSBAn ⊂ LSBAn+1 induced by {x1, … , xn} ⊂ { x1, … , xn, xn+1}.
The atoms of LSBAn are no longer atomic in LSBAn+1. The left-handed identity x = (x∧y) + (x\y)
gives the following “subatomic” decomposition of the original atoms.
(x1∧x2∧…∧xk) \ (xk+1∨…∨ xn)
= (x1∧x2∧…∧xk∧xn+1) \ (xk+1∨…∨ xn) + (x1∧x2∧…∧xk)\ (xk+1∨…∨ xn+1)
Both components of the new decomposition are atoms in LSBAn+1. If say
(x1∧x2∧x3)\(x4∨…∨xn) = (x1∧x3∧x2)\(x4∨…∨xn)
in LSBAn, then their corresponding pairs of atomic components in LSBAn+1 remain equal. But if
say (x1∧x2)\(x3∨…∨xn) ≠ (x2∧x1)\(x3∨x4∨…∨xn) in LSBAn, then both corresponding pairs of
components are likewise unequal in LSBAn+1. One thus has extended the decomposition where,
while a given element remains the same, its atomic decomposition doubles in length as each new
generator is added. Thus given u ∈ LSBAn with atomic decomposition u = a1 +…+ ar in
LSBAn, each atom ak splits as bk + ck in LSBAn+1, where bi = ai∧xn+1 and ci = ai\xn+1, to give a
revised atomic decomposition u = b1 + c1 + … + br + cr in LSBAn+1. Given the uniqueness of
atomic decompositions (to within commutativity) of elements in LSBAn or in LSBAn+1, we have:
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xʺ, yʺ ∈S/R. Since S/L and S/R have finite intersections, xʹ∩yʹ and xʺ∩yʺ exist in S/L and S/R,
respectively. If xʹ∩yʹ and xʺ∩yʺ share a common image in S/D, then (xʹ, xʺ) ∩ (yʹ, yʺ) is just
(xʹ∩yʹ, xʺ∩yʺ). In general, let u0∧v0 be the meet in S/D of the respective images u0 of xʹ∩yʹ and
v0 of xʺ∩yʺ in S/D. In the respective D classes of S/L and S/R indexed by u0∧v0, unique
elements wʹ and wʺ exist (by normality) such that both xʹ∩yʹ ≥ wʹ in S/L and xʺ∩yʺ ≥ wʺ in S/R.
The intersection (xʹ, xʺ) ∩ (yʹ, yʺ) is then precisely (wʹ, wʺ). £
The practical consequence of all this as follows: just as all skew Boolean algebras can be
constructed in principle from pairs of left- and right-handed skew Boolean algebras (SL, SR)
having a common maximal commutative Boolean image B using the fibered product, SL ×B SR, so
also all skew Boolean ∩-algebras can be constructed from pairs of left- and right-handed skew
Boolean ∩-algebras (SL, SR) with a common maximal generalized Boolean algebra image B
(with SL and SR viewed as skew Boolean algebras) by exactly the same process, SL ×B SR, even
when their corresponding ∩-outcomes may have distinct locations relative to B. It follows that
the study of skew Boolean ∩-algebras can, in principle, be reduced to studying right-handed skew
Boolean ∩-algebras, or their term-equivalent left-handed duals, since pairs sharing common
maximal lattice images can be spliced together at will. All this is illustrated in the following case
of infinite free skew Boolean algebras.
Atom splitting and the case of infinite free skew Boolean algebras
Consider the inclusion LSBAn ⊂ LSBAn+1 induced by {x1, … , xn} ⊂ { x1, … , xn, xn+1}.
The atoms of LSBAn are no longer atomic in LSBAn+1. The left-handed identity x = (x∧y) + (x\y)
gives the following “subatomic” decomposition of the original atoms.
(x1∧x2∧…∧xk) \ (xk+1∨…∨ xn)
= (x1∧x2∧…∧xk∧xn+1) \ (xk+1∨…∨ xn) + (x1∧x2∧…∧xk)\ (xk+1∨…∨ xn+1)
Both components of the new decomposition are atoms in LSBAn+1. If say
(x1∧x2∧x3)\(x4∨…∨xn) = (x1∧x3∧x2)\(x4∨…∨xn)
in LSBAn, then their corresponding pairs of atomic components in LSBAn+1 remain equal. But if
say (x1∧x2)\(x3∨…∨xn) ≠ (x2∧x1)\(x3∨x4∨…∨xn) in LSBAn, then both corresponding pairs of
components are likewise unequal in LSBAn+1. One thus has extended the decomposition where,
while a given element remains the same, its atomic decomposition doubles in length as each new
generator is added. Thus given u ∈ LSBAn with atomic decomposition u = a1 +…+ ar in
LSBAn, each atom ak splits as bk + ck in LSBAn+1, where bi = ai∧xn+1 and ci = ai\xn+1, to give a
revised atomic decomposition u = b1 + c1 + … + br + cr in LSBAn+1. Given the uniqueness of
atomic decompositions (to within commutativity) of elements in LSBAn or in LSBAn+1, we have:
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