All Questions
Tagged with ra.rings-and-algebras universal-algebra
13 questions
2
votes
1
answer
211
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Terminology for a monoid $(H, \cdot)$ s.t. $ax=a$ or $xa =a$ only if $x$ is a unit
Let $(H, \cdot)$ be a (multiplicative) monoid. Is there any consolidated name for the following Property $\text{(P)}$, or for the class of monoids for which it is satisfied?
$$\text{(P) If }\,xy = x\...
- 10.5k
13
votes
3
answers
678
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IBN for algebraic theories
Let us say that a finitary algebraic theory $\tau$ has IBN (invariant basis number) if the free functor $F : \mathsf{Set} \to \mathsf{Mod}(\tau)$ reflects the isomorphism relation: If $S,T$ are sets ...
- 63.1k
8
votes
2
answers
596
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If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
5
votes
3
answers
542
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Congruences that aren't "finite from above"
Let $\mathfrak{A}=(A;...)$ be an algebra in the sense of universal algebra. Say that a congruence $\sim$ on $\mathfrak{A}$ is parafinite iff there is an equivalence relation $E\subseteq A^2$ with ...
- 21.2k
62
votes
5
answers
10k
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Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)
Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated?
Basically I want to believe I can ...
- 11.2k
22
votes
3
answers
6k
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Why are ring actions much harder to find than group actions?
I admit freely that the following question is a bit of a fishing expedition inspired by this lovely "definition" of a module as found on Wikipedia:
A module is a ring action on an abelian group.
...
- 15.5k
11
votes
3
answers
939
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What is the smallest variety of algebras containing all fields?
A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
- 2,464
9
votes
1
answer
712
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Generalizing detropicalization
Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...
- 19.7k
9
votes
2
answers
661
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Birkhoff's completeness theorem put into practice
Birkhoff's completeness theorem (see here, Theorem 14.19) states that an equation which is true in all models of an algebraic theory can be proven in equational logic.
Question. Does the proof of ...
- 63.1k
7
votes
2
answers
488
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Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
- 10.5k
5
votes
2
answers
974
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Shape of axioms in algebraic structures
When defining algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in many ...
- 1,341
4
votes
1
answer
1k
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Commutative associative rational binary operations
What are all the nondegenerate rational binary operations that are commutative and associative? (Examples: $(x,y) \mapsto x+y$, $xy+x+y$, $xy/(x+y)$.)
Feel free to re-tag if you can think of ...
- 19.7k
3
votes
1
answer
328
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What does it mean for the surreal numbers/partizan games to be "universally embedding"?
In "On numbers and games", Conway writes that the surreal Numbers form a universally embedding totally ordered Field. Later Jacob Lurie proved that (the equivalence classes of) the partizan ...
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