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• CommentRowNumber1.
• CommentAuthorTobyBartels
• CommentTimeDec 15th 2018

Relative pseudocomplements

• CommentRowNumber2.
• CommentAuthorGuest
• CommentTimeMar 5th 2021

Currently, the text under “In a lattice” has the text:

Given a lattice $L$ and two elements $x$ and $y$ of $L$, a relative complement of $x$ relative to $y$ is an element $y \setminus x$ such that:

• $x \wedge (y \setminus x) = \bot$ (where $\wedge$ is the meet in the lattice and $\bot$ is the bottom of the lattice) and
• $y \leq x \vee (y \setminus x)$ (where $\vee$ is the join in the lattice).

And then in the text beneath, it says.

In any case, complements are unique.

However, this isn’t true even in a Boolean algebra. Consider the power set of ${0,1,2}$ and let $x$ be ${0}$ and $y$ be ${0,1}$. $z={1,2}$ satisfies the two conditions above but is not equal to the usual set difference $y \setminus x = {1}$.

Is there a reference for (something like) this definition for relative complement?

Also, while I’m here:

The term Boolean ring is sometimes used for something like a Boolean algebra in which there may be no top element (and therefore no complements) but still relative complements; compare the notions of ring vs algebra at sigma-algebra. That is, a Boolean ring is a bounded-below distributive lattice in which all relative complements exist.

I believe these are also called “generalized Boolean algebra”, see e.g. Wikipedia though I think the terminology goes back to Stone. This Wikipedia article also defines the relative complement via a pair of conditions:

Defining a ∖ b as the unique x such that (a ∧ b) ∨ x = a and (a ∧ b) ∧ x = 0,

• CommentRowNumber3.
• CommentAuthorj.c.
• CommentTimeMar 5th 2021
(Oops, I didn't mean to post that entirely anonymously; I somehow got signed out while posting.)