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nLab > Latest Changes: BoolAlg

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    • CommentRowNumber1.
    • CommentAuthorJohn Baez
    • CommentTimeMay 21st 2023

    Explained why the square of the finite power set functor is (or “can be made into”) the monad for Boolean algebras.

    diff, v14, current

    • CommentRowNumber2.
    • CommentAuthorJohn Baez
    • CommentTimeMay 22nd 2023

    I corrected my description of the monad for boolean algebras.

    diff, v16, current

    • CommentRowNumber3.
    • CommentAuthorPause706
    • CommentTimeDec 5th 2024
    The part about morphisms between Boolean algebras seems odd to me
    " BoolAlg is the category whose objects are boolean algebras and whose morphisms are lattice homomorphisms, that is functions which preserve finitary meets and joins (equivalently, binary meets and joins and the top and bottom elements); it follows that the homomorphisms preserve negation. " From the first paragraph of description.

    I don't get how they are equivalent; in particular why lattice homomorphisms between boolean algebras must preserve top and bottom. It only preserves them on the image (I think this follows from 1 v a = 1 and 0 v a = a both being positive formulas) and a counterexample would be the map from the trivial bool alg with 1 element to the 2 element boolean algebra, clearly this does not preserve bottom and top. Another less trivial counter, the 2 element boolean alg maps into the square lattice, if we pick the top and 1 element directly beneath that of the latter, we get a lattice homo not preserving alg homos.

    Also I don't get why we specify binary and finitary meets since they should be equivalent via induction. f(avbvc) = f(avb) v f(c) = f(a) v f(b) v f(c).

    Thanks for reading

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