Page 217 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press

VI: Skew Lattices in Rings

Proof. In light of Theorem 6.1.3, the assertion about Se is obvious. In general, for all x ∈ Re,
x = e + ea(1 – e) = e + ea(1 – e)f + ea(1 – e)(1 – f )

for some a in R (x itself will do). Thus xf = ef + ea(1 – e)f, so that xf ∈Ref only when
ea(1 – e)f = efb(1 – ef) for some b in R which occurs precisely when ea(1 – e)f = efb(1 – e)f for
some b since (1 – e)f = (1 – ef)f. Thus,

S(e⎮f) ⊆ e + efRf(1– e) + eR(1– e)(1– f) ⊆ Re.
Since

{e + efRf(1– e) + eR(1– e)(1– f)}f = ef + efRf(1– e) = ef + efRf(1 – ef) ⊆ Ref,

the reverse inclusion holds. Case (ii) follows immediately from the general case. For case (i), if
f > e, then S(e⎮f) = e + eRf(1– e) + eR(1– e)(1– f) which reduces to e + eR(1– e) = Re. If e > f,
then both f(1 – e) = 0 and (1– e)(1– f) = 1 – e, so that S(e⎮f) = e + eR(1– e) = Re again. £

The general case and the two special cases are illustrated in the following block matrix
diagrams where u, x, y and z hold arbitrary values.

⎡1 0 0 0⎤ ⎡0 0 0 0⎤ ⎡1 0 0 y⎤
⎢ ⎥
e= ⎢ 0 1 0 0⎥ f= ⎢ 0 1 0 0⎥ S(e⎮f) = all ⎢ 0 1 x z ⎥ .
⎢ 0 0 0 0⎥ ⎢ 0 0 1 0⎥ ⎣⎢ 0 0 0 0
⎣⎢ 0 0 0 0 ⎦⎥ ⎣⎢ 0 0 0 0 ⎥⎦ 0 0 0 0 ⎥⎦

⎡1 0 0 0⎤ ⎡0 0 0 0⎤ ⎡1 0 u y⎤
⎢ ⎥
e= ⎢ 0 1 0 0⎥ f= ⎢ 0 1 0 0⎥ S(e⎮f) = all ⎢ 0 1 x z ⎥ .
⎢ 0 0 0 0⎥ ⎢ 0 0 0 ⎥ ⎢⎣ 0 0 0 0
⎢⎣ 0 0 0 0 ⎦⎥ ⎣⎢ 0 0 0 0 ⎥⎦ 0 0 0 0 ⎥⎦
0

⎡1 0 0 0⎤ ⎡0 0 0 0⎤ ⎡1 0 0 y⎤
⎢ 0⎥ ⎢ ⎥
e= ⎢ 0 1 0 0⎥ f= ⎢ 0 0 0 ⎥ S(e⎮f) = all ⎢ 0 1 0 z ⎥ .
⎢ 0 0 0 0⎥ ⎢⎣ 0 0 1 0 ⎦⎥ ⎢⎣ 0 0 0 0
⎣⎢ 0 0 0 0 ⎦⎥ 0 0 0 0 0 0 0 0 ⎦⎥

The case of matrix rings over fields

Given a matrix ring R = Mn(F) over a (possibly skew) field F, let Δ denote the maximal
lattice of 0-1 diagonal matrices and let e1, … , en denote the atoms of Δ. Thus ei is the matrix
with 1 in the ii-position and 0 elsewhere. The support of any e ∈ Δ is supp(e) = {i ∈ *⎮eii = 1}.

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