Page 58 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press

athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond

⎡ 0 A AB ⎤ ⎡0 C CD⎤ ⎡0 C AB + CD − AD⎤ ⎡0 C CB⎤
⎢ 0 I B ⎥ ○ ⎢0 I D ⎥ = ⎢0 I B ⎥ ≠ ⎢0 I B ⎥
⎢⎣ 0 0 0 ⎦⎥ ⎢⎣0 0 0 ⎥⎦ ⎢⎣0 0 0 ⎦⎥ ⎣⎢0 0 0 ⎦⎥

⎡0 C AB + CD − AD⎤2 ⎡0 C CB⎤
except in special cases. However, ⎢0 I B ⎥ = ⎢0 I B ⎥ . £
⎣⎢0 0 ⎥ ⎣⎢0 0 0 ⎦⎥
0 ⎦

This leads us to define a cubic join e∇f by

e∇f = e + f + fe – efe – fef.

The cubic join extends the quadratic join given by ○ in the following sense:

1) If e, f, ef, fe ∈ E(R), then (e ○ f)2 = e + f + fe – efe – fef = e∇f ∈ E(R) also.
2) Every skew lattice (S; ○, •) in a ring is trivially a skew lattice under ∇ and •,
since e∇f reduces to e ○ f whenever the latter is idempotent.
3) Situations occur where e ○ f is not idempotent but e∇f is. Indeed, e∇f ∈ E(S)
whenever e, f, ef, fe ∈ E(S). (This combines (1) with the previous example.)
4) Caveat: While ○ is always associative, ∇ needn’t be, even when giving
idempotent closure.

Example 2.3.2. (Karin Cvetko-Vah) Consider the following matrix band with two D-
classes:

⎧⎡0 x1 x2 x1y1 + x2 y2 ⎤ ⎫
⎪⎪⎪⎨⎢⎢⎢00 1 0 ⎪⎪
y1 ⎥ x1, x2 , y1, y2 ∈F⎬
0 1 y2 ⎥ ⎪
⎥
⎪⎩⎣⎢0 0 0 0 ⎥⎦
⎭⎪

↓

⎧⎡0 u uv1 uv2 ⎤ ⎫
⎪⎨⎪⎪⎢⎢⎢00 1 v1 ⎥ ∈F ⎬⎪⎪
0 0 v2 ⎥ u, v1, v2
0 ⎥ ⎪

⎩⎪⎣⎢0 0 0 0 ⎦⎥ ⎭⎪

This band is also closed under ∇. However, upon setting

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