Page 201 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press

V: Further Topics in Skew Lattices

b1 – b2 – d2 – d1 C– b3 – b4 – d4 – d3 C– b5  b2n –1 – b2n – d2n – d2n –1 – (b1)

A C A A C A A C A C

or for n = ω,

 d–3 – b–1 – b0 C– d0 – d–1 – b1 – b2 – d2 – d1 – b3 – b4 – d4  .

C A A C A C A C A C

We denote the left-handed skew lattice thus determined by Un and its right-handed dual by Vn.
Our example is U2. All Un and Vn for n ≥ 2 are nondistributive since A-cosets and C-cosets in the
single component Bn for n ≥ 2 need not have nonempty intersection. Must a categorical skew
lattice containing no copies of Un or Vn, be linearly distributive? Here is a counterexample.

Example 5.6.6: The underlying set of S is U2 × &2.

(a1, 0) – (a1,1) – (a2 , 0) – (a2 ,1)
    ...    

(b1, 0) – (b1,1) A– (b2 , 0) – (b2 ,1) – ... – (d3 , 0 ) A– (d3, 1) A– (d4 , 0) A– (d4 ,1)
A A

    ...    

(c1, 0) – (c1,1) – (c2 , 0) – (c2 ,1)

All cosets are just cartesian products of U2-cosets with &2. The coset bijections are as follows.

1) (a1, 0) – (a1,1) – (a2 , 0) – (a2 ,1) → (b1, 0) – (b1,1) – (b2 , 0) – (b2 ,1)
2) (a1, 0) – (a1,1) – (a2 , 0) – (a2 ,1) → (b3, 0) – (b3,1) – (b4 , 0) – (b4 ,1)
3) (a1, 0) – (a1,1) – (a2 , 0) – (a2 ,1) → (d1, 0) – (d1,1) – (d2 , 0) – (d2 ,1)
4) (a1, 0) – (a1,1) – (a2 , 0) – (a2 ,1) → (d3, 0) – (d3,1) – (d4 , 0) – (d4 ,1)
5) (b1, 0) – (b1,1) – (d3, 0) – (d3,1) → (c1, 0) – (c1,1) – (c2 , 0) – (c2 ,1)
6) (b2 , 0) – (b2 ,1) – (d2 , 0) – (d2 ,1) → (c1, 0) – (c1,1) – (c2 ,1) – (c2 , 0)
7) (b3, 0) – (b3,1) – (d1, 0) – (d1,1) → (c1, 0) – (c1,1) – (c2 , 0) – (c2 ,1)
8) (b4 , 0) – (b4 ,1) – (d4 , 0) – (d4 ,1) → (c1, 0) – (c1,1) – (c2 ,1) – (c2 , 0)

For each bijection, the outcomes occur in the same order as the inputs are listed. These coset
bijections collectively determine operations ∨ and ∧ on S making S a left-handed categorical

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