Page 47 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press

II: Skew Lattices

{1}

¢¢¢
  
¢¢



¢

↓
{0}

Such cases naturally raise the question as to whether a counter-operation ∨ exists such

that the join class De ∨ Df of two D-classes, De and Df, is given as De∨f. This leads to a simple
but important observation:

Theorem 2.1.7. Any multiplicative band B in a ring that is also closed under the circle
operation x ○ y = x + y – xy has the following properties:

1. B is also a band under ○.
2. x(x ○ y) = x = (y ○ x)x and x ○ (xy) = x = (yx) ○ x hold on B:

Thus (B; ○, •) is a skew lattice. In this case the elements in B also satisfy the
identity:
3. xyx + yxy = xy + yx.

Proof. To see (1), note that ○ is always associative and x○x = x + x – x2 = x if and only if xx = x.
Thus if B is closed under ○, then (B, ○) is indeed a band. (2) is straightforward. For instance,

x(x ○ y) = xx + xy – xxy = x + xy – xy = x and x ○ (xy) = x + xy – xxy = x + xy – xy = x.

Given the assumptions, x + y – xy = x ○ y = (x ○ y)2 = (x + y – xy)2 = x + yx + y – yxy – xyx, and
(3) follows. £

The terms in (3) form a rectangular subalgebra for which the diagonal sums are equal.

xyx R xy
L L
yx R yxy

Two classes of bands closed under ○ are given in the next theorem. But first a lemma:

Lemma 2.1.8. Let B be a right regular band in a ring R and let e, f ∈ B. Then:

1) e + f – ef ∈ E(R), and

2) B  {e + f – ef } generates a right regular band.

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