Page 148 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 148
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Lemma 4.4.15 Given u be an atom of LSBAn and let a = b + c be the atomic decompo-
sition of a in LSBAn+1 where b = a∧xn+1 and c = a\xn+1. Then the following are equivalent:
i) u ≥ a in LSBAn (and thus in LSBAn+1).
ii) u ≥ b in LSBAn+1.
iii) u ≥ c in LSBAn+1. !
Comments. (1) Thus a is in the atomic decomposition of u in LSBAn iff b [or c and hence both]
is in the atomic decomposition of u in LSBAn+1.
(2) LSBAn+1 has n+1 “natural” copies of LSBAn in it, each generated by one of n+1
subsets of {x1, … , xn+1} of size n. Likewise ⎛ n + 1⎞ natural copies of LSBAn–1 lie in LSBAn+1,
⎝⎜ n − 1⎟⎠
etc.
This leads us to infinite free algebras with necessarily infinite generating sets. If X is
infinite, then LSBAX is the upward directed union of its finite free subalgebras:
LSBAX = ∪{LSBAY⎮∅ ≠ Y ⊆ X & |Y| < ∞}.
Given u and v of LSBAX, each occurs in some finite free subalgebra, say u in LSBAY and v in
LSBAZ for finite subsets Y and Z of X. Thus u∧v, u∨v and u\v are calculated in the larger finite
subalgebra LSBAY∪Z or in any finite LSBAW where Y∪Z ⊆ W. Of course calculations of u∧v,
u∨v and u\v do not change in passing from LSBAY∪Z to any properly larger LSBAW. What
changes is their atomic decompositions; such changes, however, are derived from the original
decompositions in LSBAY∪Z by (possibly repeated) atomic splitting. Ultimately in LSBAX for X
infinite, no atoms exist. (If a is an atom, then it appears as such in LSBAY for some finite Y; but it
immediately looses its atomic status in a properly larger free subalgebra.) Atoms are only
relevant in its finite subalgebras. This is a fundamental difference between finite and infinite free
algebras. Another fundamental difference is as follows.
Recall that the center of a skew lattice, consisting of elements that both ∧-commute and
∨-commute with all elements, is the union of all singleton D-classes. In LSBAn (or RSBAn or
SBAn) it is the set of all n atoms of the form x1\(x2∨…∨xn) and the subalgebra they generate
consisting of all orthosums of such atoms. But, except for 0, none of these orthosums remain
central in LSBAn+1. For each atom,
xn+1∧(x1\(x2∨…∨xn)) = ((xn+1∧x1)\(x2∨…∨xn))
≠ ((x1∧ xn+1)\(x2∨…∨xn)) = (x1\(x2∨…∨xn))∧ xn+1
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Lemma 4.4.15 Given u be an atom of LSBAn and let a = b + c be the atomic decompo-
sition of a in LSBAn+1 where b = a∧xn+1 and c = a\xn+1. Then the following are equivalent:
i) u ≥ a in LSBAn (and thus in LSBAn+1).
ii) u ≥ b in LSBAn+1.
iii) u ≥ c in LSBAn+1. !
Comments. (1) Thus a is in the atomic decomposition of u in LSBAn iff b [or c and hence both]
is in the atomic decomposition of u in LSBAn+1.
(2) LSBAn+1 has n+1 “natural” copies of LSBAn in it, each generated by one of n+1
subsets of {x1, … , xn+1} of size n. Likewise ⎛ n + 1⎞ natural copies of LSBAn–1 lie in LSBAn+1,
⎝⎜ n − 1⎟⎠
etc.
This leads us to infinite free algebras with necessarily infinite generating sets. If X is
infinite, then LSBAX is the upward directed union of its finite free subalgebras:
LSBAX = ∪{LSBAY⎮∅ ≠ Y ⊆ X & |Y| < ∞}.
Given u and v of LSBAX, each occurs in some finite free subalgebra, say u in LSBAY and v in
LSBAZ for finite subsets Y and Z of X. Thus u∧v, u∨v and u\v are calculated in the larger finite
subalgebra LSBAY∪Z or in any finite LSBAW where Y∪Z ⊆ W. Of course calculations of u∧v,
u∨v and u\v do not change in passing from LSBAY∪Z to any properly larger LSBAW. What
changes is their atomic decompositions; such changes, however, are derived from the original
decompositions in LSBAY∪Z by (possibly repeated) atomic splitting. Ultimately in LSBAX for X
infinite, no atoms exist. (If a is an atom, then it appears as such in LSBAY for some finite Y; but it
immediately looses its atomic status in a properly larger free subalgebra.) Atoms are only
relevant in its finite subalgebras. This is a fundamental difference between finite and infinite free
algebras. Another fundamental difference is as follows.
Recall that the center of a skew lattice, consisting of elements that both ∧-commute and
∨-commute with all elements, is the union of all singleton D-classes. In LSBAn (or RSBAn or
SBAn) it is the set of all n atoms of the form x1\(x2∨…∨xn) and the subalgebra they generate
consisting of all orthosums of such atoms. But, except for 0, none of these orthosums remain
central in LSBAn+1. For each atom,
xn+1∧(x1\(x2∨…∨xn)) = ((xn+1∧x1)\(x2∨…∨xn))
≠ ((x1∧ xn+1)\(x2∨…∨xn)) = (x1\(x2∨…∨xn))∧ xn+1
146
