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nLab > Latest Changes: Boolean algebra

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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 12th 2012

    Added more material to Boolean algebra, particularly the principle of duality and the connection to Boolean rings, and a wee bit of material on Stone duality.

    Stone duality deserves greater expansion, bringing out the dualities via ambimorphic (ahem, schizophrenic) structures on the 2-element set, and mentioning the connection to Chu spaces. Another day, another dollar.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeMar 12th 2012

    You might want to look also at Boolean ring.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 24th 2024
    • CommentRowNumber4.
    • CommentAuthorJ-B Vienney
    • CommentTimeJul 31st 2024

    Added that a Boolean algebra can be defined as a cartesian *-autonomous poset.

    diff, v26, current

    • CommentRowNumber5.
    • CommentAuthorPause706
    • CommentTimeDec 5th 2024
    In the article on boolean algebra (and a similar one in complete boolean algebras), the statement "Any lattice homomorphism automatically preserves ¬"
    But that doesn't sound right, how do you know what happens to bottom (0) and top (1) after a lattice homomorphism? It should only preserve it for surjective homos (Lyndon; clearly a v 1 = 1 and a v 0 = a are both positive formulas). A counter example would be the 2 element boolean alg to the square boolean alg mapping top _A to top_B and bottom_A to anything other than the bottom_B. A lattice homo but not an Boolean alg homo and if it doesn't preserve top/bottoms, it clearly can't preserve complements. If lattice homos must preserve top/bottom then it makes more sense but why would that be true? Shouldn't we say bounded lattice homos instead? Perhaps change it to "BoolAlg is a full subcategory of BouLat, the category of bounded lattice with morphisms as the homos in universal algebra" And distributivity clearly follows since homos give "operation properties" for free, then establish uniqueness of complements etc.

    Thanks for reading :)
    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 12th 2024

    Re #5:

    If lattice homos must preserve top/bottom then it makes more sense but why would that be true? Shouldn’t we say bounded lattice homos instead?

    “Lattice” means “bounded lattice”, not “pseudolattice” in this article and other articles.

    For whatever reasons, nonunital structures (e.g., rings without identities or lattices without top/bottom elements) used to be much more popular in mid-20th century.

    But such conventions appear to be rare these days. For example, “ring” always means “unital ring” in all 21st century articles I have seen. (C*-algebras appear to be a sole outlier.)

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