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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeDec 15th 2018

    Relative pseudocomplements

    diff, v4, current

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTimeMar 5th 2021

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

    Given a lattice LL and two elements xx and yy of LL, a relative complement of xx relative to yy is an element yxy \setminus x such that:

    • x(yx)= x \wedge (y \setminus x) = \bot (where \wedge is the meet in the lattice and \bot is the bottom of the lattice) and
    • yx(yx) 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{0,1,2} and let xx be 0{0} and yy be 0,1{0,1}. z=1,2z={1,2} satisfies the two conditions above but is not equal to the usual set difference yx=1y \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.)
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