Page 200 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
∧ ai b j bk d j dk c j ∨ ai b j bk d j dk c1 c2
a1 ai a j ak –2 a j dk –2 a1 a1
a1 a1 b1 b3 d1 d3 c j a2 ai a j ak –2 a j dk –2 a2 a2
b1 ai b j bk d j dk b1 d3
a2 a2 b2 b4 d2 d4 c j b2 ai b j bk d j dk b2 d2
b3 ai b j bk d j dk b3 d1
b1 b1 b1 b1 b1 b1 c1
b2 b2 b2 b2 b2 b2 c1 b4 ai b j bk d j dk b4 d4 .
b3 b3 b3 b3 b3 b3 c1
b4 b4 b4 b4 b4 b4 c1 d1 ai b j bk d j dk b3 d1
d1 d1 d1 d1 d1 d1 c2 d2 ai b j bk d j dk b2 d2
d2 d2 d2 d2 d2 d2 c2 d3 ai b j bk d j dk b1 d3
d3 d3 d3 d3 d3 d3 c2 d4 ai b j bk d j dk b4 d4
d4 d4 d4 d4 d4 d4 c2 c1 ai b j bk d j dk c1 c2
c1 c1 c1 c1 c1 c1 c1 c2 ai b j bk d j dk c1 c2
c2 c2 c2 c2 c2 c2 c2
Their common Hasse diagram is as follows, with bi –C dj iff i + j ≡ 0 (mod 4).
A a1 – a2
B b1 A– b2 – b3 – b4 – d1 – d2 – d3 A– d4
A A
C
c1 – c2
In both cases, a1 > bodd, dodd and a2 > beven, deven, all bi > c1, all di > c2, and a1, a2 > both c1, c2.
Thus both skew chains are categorical since all cosets involving just A and C are trivial. We
denote the left-handed skew lattice thus determined by U and its right-handed dual by V. Both U
and V are not distributive. Indeed, given the coset structure on B, we get a1 ∧ (b2 ∨ c2) = a1 ∧ d2
= d1, while (a1 ∧ b2) ∨ (a1 ∧ c2) = b1 ∨ c2 = d3 ≠ d1 in U. V is handled similarly. Note that in
both U and V, B is a AC-connected, but a1 > b1, b3 > c1, and also a2 > b2, b4 > c1, etc.
This example is the n = 2 case of a sequence of similar skew chains A > Bn > C, where A
= {a1, a2} and C = {c1, c2} as above, but Bn = {b1, b2, b3, … , b2n} ∪ {d1, d2, d3, … , d2n} for n
finite or{… , b–2, b–1, b0, b1, b2, …} ∪ {… , d–2, d–1, d0, d1, d2, …} for n = ω. In all cases,
a1 > bodd, dodd; a2 > beven, deven; all bi > c1; all di > c2; and a1, a2 > both c1, c2.
(That a1, a2 > both c1, c2 insures that these skew chains are categorical since all “outer” cosets
involving just A and C are trivial.)
An important aspect of these examples is the fact that all A-cosets and C-cosets in Bn are
of size 2 and are interconnected in the following cyclic fashion in the finite case,
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∧ ai b j bk d j dk c j ∨ ai b j bk d j dk c1 c2
a1 ai a j ak –2 a j dk –2 a1 a1
a1 a1 b1 b3 d1 d3 c j a2 ai a j ak –2 a j dk –2 a2 a2
b1 ai b j bk d j dk b1 d3
a2 a2 b2 b4 d2 d4 c j b2 ai b j bk d j dk b2 d2
b3 ai b j bk d j dk b3 d1
b1 b1 b1 b1 b1 b1 c1
b2 b2 b2 b2 b2 b2 c1 b4 ai b j bk d j dk b4 d4 .
b3 b3 b3 b3 b3 b3 c1
b4 b4 b4 b4 b4 b4 c1 d1 ai b j bk d j dk b3 d1
d1 d1 d1 d1 d1 d1 c2 d2 ai b j bk d j dk b2 d2
d2 d2 d2 d2 d2 d2 c2 d3 ai b j bk d j dk b1 d3
d3 d3 d3 d3 d3 d3 c2 d4 ai b j bk d j dk b4 d4
d4 d4 d4 d4 d4 d4 c2 c1 ai b j bk d j dk c1 c2
c1 c1 c1 c1 c1 c1 c1 c2 ai b j bk d j dk c1 c2
c2 c2 c2 c2 c2 c2 c2
Their common Hasse diagram is as follows, with bi –C dj iff i + j ≡ 0 (mod 4).
A a1 – a2
B b1 A– b2 – b3 – b4 – d1 – d2 – d3 A– d4
A A
C
c1 – c2
In both cases, a1 > bodd, dodd and a2 > beven, deven, all bi > c1, all di > c2, and a1, a2 > both c1, c2.
Thus both skew chains are categorical since all cosets involving just A and C are trivial. We
denote the left-handed skew lattice thus determined by U and its right-handed dual by V. Both U
and V are not distributive. Indeed, given the coset structure on B, we get a1 ∧ (b2 ∨ c2) = a1 ∧ d2
= d1, while (a1 ∧ b2) ∨ (a1 ∧ c2) = b1 ∨ c2 = d3 ≠ d1 in U. V is handled similarly. Note that in
both U and V, B is a AC-connected, but a1 > b1, b3 > c1, and also a2 > b2, b4 > c1, etc.
This example is the n = 2 case of a sequence of similar skew chains A > Bn > C, where A
= {a1, a2} and C = {c1, c2} as above, but Bn = {b1, b2, b3, … , b2n} ∪ {d1, d2, d3, … , d2n} for n
finite or{… , b–2, b–1, b0, b1, b2, …} ∪ {… , d–2, d–1, d0, d1, d2, …} for n = ω. In all cases,
a1 > bodd, dodd; a2 > beven, deven; all bi > c1; all di > c2; and a1, a2 > both c1, c2.
(That a1, a2 > both c1, c2 insures that these skew chains are categorical since all “outer” cosets
involving just A and C are trivial.)
An important aspect of these examples is the fact that all A-cosets and C-cosets in Bn are
of size 2 and are interconnected in the following cyclic fashion in the finite case,
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