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Results tagged with ra.rings-and-algebras
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user 8628
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
6
votes
2
answers
233
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Realizing a monoid as $\mathrm{End}(G)$ for some graph $G$
This question relates to Realizing groups as automorphism groups of graphs.
Given a monoid $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to $M$?
- 48.9k
6
votes
2
answers
379
views
Irreducible elements in endomorphism rings
Let $(G, +)$ be a commutative group. The endomorphism set $\text{End}(G)$ of all group endomorphisms $f:G\to G$ is a ring, where $+$ is taken pointwise and the multiplication is the composition of end …
- 48.9k
4
votes
1
answer
378
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Finite distributive lattices as lattice of ideals of a finite ring
Is there a finite distributive lattice that is not isomorphic to the lattice of ideals of a finite ring?
- 48.9k
4
votes
1
answer
439
views
Non-associative commutative "group"
When dealing with some hash functions that I was trying to speed up, I toyed with a binary operation with the goal to "approximate" the addition on $\{0,1\}^*$ when seen as binary representation of th …
- 48.9k
4
votes
3
answers
513
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Non-principal prime ideals in infinite distributive lattices
Given an infinite distributive lattice $L$, does $L$ contain a non-principal prime ideal $I$, or a non-principal prime filter $F$? ($I$ is said to be principal if there is $x\in L$ such that $I=\{y\in …
- 48.9k
3
votes
Non-principal prime ideals in infinite distributive lattices
A counterexample is $\omega+1$, turned upside down. However, it is true that every infinite distributive lattice contains either a non-principal prime ideal or a non-principal prime filter. This is du …
- 48.9k
2
votes
1
answer
80
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Ideal colorings of rings
This is an update of an older question, suggested in this comment by Zach Teitler.
Let $R$ be a ring with more than 1 element, and let $A$ be a non-empty set. We call a map $c:R\to A$ an ideal colori …
- 48.9k
2
votes
0
answers
149
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Binary operation approximating "addition" on $2^\omega$
Motivation. In computer science, addition of integers $a+b$ can be approximated by a very fast operation: $(a,b)\mapsto (a\oplus b) \oplus ((a\land b) \ll 1)$, where $\oplus$ denotes bitwise XOR, $\la …
- 48.9k
1
vote
1
answer
120
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"Coloring" the ideals of a ring
Let $R$ be a ring with more than 1 element, and let $A$ be a non-empty set. We call a map $c:R\to A$ an ideal coloring if for every nonempty ideal $I$ with $I\neq\{0\}$ the restriction $c|_I$ is not c …
- 48.9k
-1
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Structure of $Hom(L_1,L_2)$, where $L_i$ are distributive lattices
Something more general holds: if $P$ is a poset and $L$ is a lattice, then $\text{Hom}(P,L)$ is also a lattice. If $f,g \in \text{Hom}(P,L)$ we define $f\vee g: P\to L$ by $$(f\vee g)(x) = f(x) \vee g …
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