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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeDec 15th 2018
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexRelative pseudocomplements <a href="https://ncatlab.org/nlab/revision/diff/relative+complement/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/relative+complement/4">v4</a>, <a href="https://ncatlab.org/nlab/show/relative+complement">current</a>

   Relative pseudocomplements

   diff, v4, current

2. • CommentRowNumber2.
   • CommentAuthorGuest
   • CommentTimeMar 5th 2021
   • PermaLink
   Author: Guest
   Format: MarkdownItexCurrently, 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 lattice|bounded-below]] [[distributive lattice]] in which all relative complements exist. I believe these are also called "generalized Boolean algebra", see e.g. [Wikipedia](https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations) though I think the terminology goes back to [Stone](https://www.jstor.org/stable/2371008?seq=1). 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,

   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,

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

   (Oops, I didn't mean to post that entirely anonymously; I somehow got signed out while posting.)