# Questions tagged [boolean-algebras]

A Boolean algebra is a commutative ring satisfying x²=x for every x, and sometimes required to have a unit; they have characteristic 2. For coding theory (notably dealing with subsets linear subspaces of spaces of Boolean functions), rather use the [coding-theory] or [linear-algebra] tag.

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### Can any poset of cardinality $\leq 2^{\aleph_0}$ be embedded in ${\cal P}(\omega)/(\text{fin})$?

We endow ${\cal P}(\omega)$ with an equivalence relation by saying that $A\simeq_{\text{fin}} B$ iff the symmetric difference $A\Delta B$ is finite. The resulting set of equivalence classes is denoted ...
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### An algebra with two multiplications, based on series-parallel diagrams?

Here is a commutative, unital, associative algebra $\mathcal{F}$ with two ways to multiply. The multiplications come from a construction with Boolean operations and series-parallel diagrams. I want ...
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### When are two forcing posets "the same"?

Let $B$ and $C$ be complete Boolean algebras. To avoid triviality I may also want them to be atomless. For $b\in B$ nonzero, denote $B\upharpoonright b=\{p\in B:p\leq b\}$, which can be viewed as a ...
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### Formulas that are valid simultaneously in a power set Boolean algebra $B$ and the 2-element Boolean algebra $\mathbf2$ [duplicate]

Note 1. Early I posted a related question Set-theoretic tautologies. But the answer did not contain any concrete references to the literature. So I posted this, more precisely formulated question, ...
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### The existence of a maximal “cross-sectional” filter on the Boolean algebra of measurable subsets of [0, 1] modulo almost everywhere equivalence

Let $\mathcal{B}([0, 1])$ be the Boolean algebra of measurable subsets of $[0, 1]$ modulo almost everywhere equivalence, i.e., two measurable sets which differ only by a Lebesgue null set are ...
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### Is every homogeneous poset a lattice?

A poset $(P,\leq)$ is homogeneous if $P\cong [a,b]$ for all $a,b\in P$ with $a<b$ (where $[a,b] := \{x\in P: a\leq x\leq b\}$). Examples of homogeneous posets include $[0,1]$, $[0,1]\cap \mathbb{Q}$...
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### Results on Boolean matrices

Matrices with entries in the finite field of two elements $\mathbb{F}_2$, and with the usual operations of matrix addition and multiplication, have been intensively studied, especially due to their ...
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### Are no infinite subsets of the set of all propositional atoms definable in this structure, even with parameters?

I asked this on Math Stack Exchange, but apparently no one paid attention to it. So, I am asking it again, filling in the background necessary to understand it. Consider a countably infinite set $P$ ...
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### Can you do boolean and of 1 and a number less than 1? [closed]

I am reading imenez, J., Echevarria, J.I., Sousa, T. and Gutierrez, D. (2012), SMAA: Enhanced Subpixel Morphological Antialiasing Computer Graphics Forum, 31: 355-364. https://doi.org/10.1111/j.1467-...
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