nForum - Discussion Feed (Boolean ring) 2023-09-25T12:10:10+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Todd_Trimble comments on "Boolean ring" (109866) https://nforum.ncatlab.org/discussion/8595/?Focus=109866#Comment_109866 2023-05-23T14:18:48+00:00 2023-09-25T12:10:10+00:00 Todd_Trimble https://nforum.ncatlab.org/account/24/ By &Fopf; 2\mathbb{F}_2-algebra, I mean an associative one. But there are many examples of what you want, of unital bilinear magma structures that are nonassociative. All you need to do is take ...

By $\mathbb{F}_2$-algebra, I mean an associative one. But there are many examples of what you want, of unital bilinear magma structures that are nonassociative.

All you need to do is take any nonassociative unital magma $M$. (For example, take $M = \{b, c, d, e\}$. and then create a multiplication table so that the first row and first column, each indexed by $e$, makes $e$ the identity element, and then fill in the rest of the multiplication table by randomly choosing entries. Most of those tables will be nonassociative.) Then, take the free vector space $\mathbb{F}_2[M]$ on $M$. The unique bilinear extension $\mathbb{F}_2[M] \times \mathbb{F}_2[M] \to \mathbb{F}_2[M]$ of the multiplication table then gives a unital bilinear magma structure which is nonassociative because $M$ is.

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Guest comments on "Boolean ring" (109864) https://nforum.ncatlab.org/discussion/8595/?Focus=109864#Comment_109864 2023-05-23T13:46:31+00:00 2023-09-25T12:10:10+00:00 Guest https://nforum.ncatlab.org/account/21/ Quick question. Are there any non-associative unital &Fopf; 2\mathbb{F}_2-algebras? Or are all unital &Fopf; 2\mathbb{F}_2-algebras associative?

Quick question. Are there any non-associative unital $\mathbb{F}_2$-algebras? Or are all unital $\mathbb{F}_2$-algebras associative?

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Todd_Trimble comments on "Boolean ring" (109862) https://nforum.ncatlab.org/discussion/8595/?Focus=109862#Comment_109862 2023-05-23T12:43:33+00:00 2023-09-25T12:10:10+00:00 Todd_Trimble https://nforum.ncatlab.org/account/24/ Done. diff, v24, current

Done.

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Todd_Trimble comments on "Boolean ring" (109861) https://nforum.ncatlab.org/discussion/8595/?Focus=109861#Comment_109861 2023-05-23T12:34:22+00:00 2023-09-25T12:10:10+00:00 Todd_Trimble https://nforum.ncatlab.org/account/24/ I’ll put something in.

I’ll put something in.

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J-B Vienney comments on "Boolean ring" (109837) https://nforum.ncatlab.org/discussion/8595/?Focus=109837#Comment_109837 2023-05-23T02:17:04+00:00 2023-09-25T12:10:10+00:00 J-B Vienney https://nforum.ncatlab.org/account/3240/ It would be nice to describe explicitly the equivalence of categories between BoolAlg and BoolRing. It is described in this entry how the two functors act on objects but not how they act on ...

It would be nice to describe explicitly the equivalence of categories between BoolAlg and BoolRing.

It is described in this entry how the two functors act on objects but not how they act on morphisms. Even less what are the natural isomorphisms $F \circ G \cong Id$ and $G \circ F \cong Id$.

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Todd_Trimble comments on "Boolean ring" (109835) https://nforum.ncatlab.org/discussion/8595/?Focus=109835#Comment_109835 2023-05-23T00:02:33+00:00 2023-09-25T12:10:10+00:00 Todd_Trimble https://nforum.ncatlab.org/account/24/ Ba-dum pah. Or: bool-yah!

Ba-dum pah. Or: bool-yah!

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Guest comments on "Boolean ring" (109833) https://nforum.ncatlab.org/discussion/8595/?Focus=109833#Comment_109833 2023-05-22T23:34:53+00:00 2023-09-25T12:10:10+00:00 Guest https://nforum.ncatlab.org/account/21/ Let boolean\mathrm{boolean} denote the type of booleans or decidable truth values. Since Booleans rings are &Fopf; 2\mathbb{F}_2-algebras, and &Fopf; 2\mathbb{F}_2 is equivalent to ...

Let $\mathrm{boolean}$ denote the type of booleans or decidable truth values. Since Booleans rings are $\mathbb{F}_2$-algebras, and $\mathbb{F}_2$ is equivalent to $\mathrm{boolean}$, one could say that Boolean rings are $\mathrm{boolean}$-algebras.

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John Baez comments on "Boolean ring" (109832) https://nforum.ncatlab.org/discussion/8595/?Focus=109832#Comment_109832 2023-05-22T22:52:21+00:00 2023-09-25T12:10:10+00:00 John Baez https://nforum.ncatlab.org/account/17/ Corrected description of the monad for Boolean rings. diff, v23, current

Corrected description of the monad for Boolean rings.

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Guest comments on "Boolean ring" (109830) https://nforum.ncatlab.org/discussion/8595/?Focus=109830#Comment_109830 2023-05-22T20:57:56+00:00 2023-09-25T12:10:10+00:00 Guest https://nforum.ncatlab.org/account/21/ added statement that one needs to use the set of all decidable subsets 2 S2^S instead of the set of all subsets &Pscr;(S)\mathcal{P}(S) in constructive mathematics to get a Boolean ring diff, ...

added statement that one needs to use the set of all decidable subsets $2^S$ instead of the set of all subsets $\mathcal{P}(S)$ in constructive mathematics to get a Boolean ring

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Todd_Trimble comments on "Boolean ring" (109821) https://nforum.ncatlab.org/discussion/8595/?Focus=109821#Comment_109821 2023-05-22T17:54:24+00:00 2023-09-25T12:10:10+00:00 Todd_Trimble https://nforum.ncatlab.org/account/24/ While not exactly wrong, I think it’s quite rare to refer to “characteristic 1”. Since “characteristic” usually refers to fields or perhaps algebras over fields, I’ve made an adjustment ...

While not exactly wrong, I think it’s quite rare to refer to “characteristic 1”. Since “characteristic” usually refers to fields or perhaps algebras over fields, I’ve made an adjustment and hope no one minds.

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Guest comments on "Boolean ring" (109819) https://nforum.ncatlab.org/discussion/8595/?Focus=109819#Comment_109819 2023-05-22T17:42:00+00:00 2023-09-25T12:10:10+00:00 Guest https://nforum.ncatlab.org/account/21/ Added related concepts section diff, v19, current

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J-B Vienney comments on "Boolean ring" (109816) https://nforum.ncatlab.org/discussion/8595/?Focus=109816#Comment_109816 2023-05-22T16:14:16+00:00 2023-09-25T12:10:10+00:00 J-B Vienney https://nforum.ncatlab.org/account/3240/ Corrected the assertion that a boolean ring is of characteristic 2 by the correct one that its characteristic divides 2, so it is 2 except when the boolean ring is trivial, in which case it is ...

Corrected the assertion that a boolean ring is of characteristic 2 by the correct one that its characteristic divides 2, so it is 2 except when the boolean ring is trivial, in which case it is 1.

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J-B Vienney comments on "Boolean ring" (109815) https://nforum.ncatlab.org/discussion/8595/?Focus=109815#Comment_109815 2023-05-22T15:38:03+00:00 2023-09-25T12:10:10+00:00 J-B Vienney https://nforum.ncatlab.org/account/3240/ Added in the idea section “The fact that the exclusive disjunction of xx and xx is the truth value “false” makes the commutative additive monoid an abelian group where &minus;x=x-x = ...

Added in the idea section “The fact that the exclusive disjunction of $x$ and $x$ is the truth value “false” makes the commutative additive monoid an abelian group where $-x = x$.”

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J-B Vienney comments on "Boolean ring" (109814) https://nforum.ncatlab.org/discussion/8595/?Focus=109814#Comment_109814 2023-05-22T15:32:55+00:00 2023-09-25T12:10:10+00:00 J-B Vienney https://nforum.ncatlab.org/account/3240/ Oh, thanks, I've learned something! So it is really ring because x (exclusive or) x = "false". Oh, thanks, I've learned something! So it is really ring because x (exclusive or) x = "false". ]]> Todd_Trimble comments on "Boolean ring" (109813) https://nforum.ncatlab.org/discussion/8595/?Focus=109813#Comment_109813 2023-05-22T15:23:11+00:00 2023-09-25T12:10:10+00:00 Todd_Trimble https://nforum.ncatlab.org/account/24/ J-B, the additive inverse in a Boolean ring does not interpret negation. (In fact, additive inversion is the identity!) Negation is given by x&map;1&minus;xx \mapsto 1 - x, or equivalently by ...

J-B, the additive inverse in a Boolean ring does not interpret negation. (In fact, additive inversion is the identity!) Negation is given by $x \mapsto 1 - x$, or equivalently by $x \mapsto 1 + x$.

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J-B Vienney comments on "Boolean ring" (109812) https://nforum.ncatlab.org/discussion/8595/?Focus=109812#Comment_109812 2023-05-22T14:58:23+00:00 2023-09-25T12:10:10+00:00 J-B Vienney https://nforum.ncatlab.org/account/3240/ Added interpretation of the connectors in the idea section. diff, v14, current

Added interpretation of the connectors in the idea section.

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John Baez comments on "Boolean ring" (109811) https://nforum.ncatlab.org/discussion/8595/?Focus=109811#Comment_109811 2023-05-22T14:15:16+00:00 2023-09-25T12:10:10+00:00 John Baez https://nforum.ncatlab.org/account/17/ Added “Idea” section. diff, v13, current

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John Baez comments on "Boolean ring" (109785) https://nforum.ncatlab.org/discussion/8595/?Focus=109785#Comment_109785 2023-05-21T20:00:11+00:00 2023-09-25T12:10:10+00:00 John Baez https://nforum.ncatlab.org/account/17/ Added a bit about the monad for Boolean rings: The free Boolean ring on a set XX can be identified with &Pscr; f&Pscr; fX\mathcal{P}_f \mathcal{P}_f X, where &Pscr; ...

The free Boolean ring on a set $X$ can be identified with $\mathcal{P}_f \mathcal{P}_f X$, where $\mathcal{P}_f \colon Set \to Set$ assigns to any set the set of all its finite subsets. In fact $\mathcal{P}_f \colon Set \to Set$ can be made into a monad in two different ways: the monad for semilattices and the monad for vector spaces over the field with 2 elements. These two monads are related by a distributive law, making $\mathcal{P}_f \mathcal{P}_f$ into the monad for Boolean rings.

Also mentioned:

The category of Boolean algebras is discussed further in BoolAlg, but some of the results about this category are proved there by working with the equivalent category of Boolean rings.

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Todd_Trimble comments on "Boolean ring" (69390) https://nforum.ncatlab.org/discussion/8595/?Focus=69390#Comment_69390 2018-06-10T04:11:26+00:00 2023-09-25T12:10:10+00:00 Todd_Trimble https://nforum.ncatlab.org/account/24/ Gave a meaning to “idempotent monoid” in concrete monoidal categories. diff, v11, current

Gave a meaning to “idempotent monoid” in concrete monoidal categories.

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Todd_Trimble comments on "Boolean ring" (69389) https://nforum.ncatlab.org/discussion/8595/?Focus=69389#Comment_69389 2018-06-10T02:10:08+00:00 2023-09-25T12:10:10+00:00 Todd_Trimble https://nforum.ncatlab.org/account/24/ There’s something just a bit odd though about the notion of idempotent monoid in AbAb. When one says that a ring is a monoid in AbAb, one means a monoid in the monoidal category ...

There’s something just a bit odd though about the notion of idempotent monoid in $Ab$. When one says that a ring is a monoid in $Ab$, one means a monoid in the monoidal category $(Ab, \otimes)$ with the standard tensor product. However, this tensor product isn’t cartesian, and so expressing the notion of an idempotent monoid (where the axiom $x x = x$ involves duplication of a variable) doesn’t go down so smoothly.

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TobyBartels comments on "Boolean ring" (69388) https://nforum.ncatlab.org/discussion/8595/?Focus=69388#Comment_69388 2018-06-10T01:57:43+00:00 2023-09-25T12:10:10+00:00 TobyBartels https://nforum.ncatlab.org/account/7/ Note that indeed any idempotent magma in AbAb is commutative. diff, v10, current

Note that indeed any idempotent magma in $Ab$ is commutative.

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