Benjamin Steinberg
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Registered User
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I am an algebraist interested in a broad range of areas. I've worked on semigroups, geometric group theory, representation theory, self-similar groups (aka automaton groups), profinite groups and random walks on semigroups and groups. I've been particularly interested in interactions between these areas and Computer Science and am fond of algorithmic questions. Recently, I've been dabbling with operator algebras associated to etale groupoids and inverse semigroups.
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18h |
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What are the main structure theorems on finitely generated commutative monoids? Probably not what you are looking for but context-free subsets of commutative monoids are semilinear, so definable in pressburger arithmetic. They have decidable membership by integer programming. In particular integer programming decides membership in submonoids so the generalized word problem is decidable. |
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18h |
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What are the main structure theorems on finitely generated commutative monoids? In fact every commutative semigroup is a semilattice of Archimedean semigroups. The Archimedean components can be strange but if you have some extra conditions they will be cancellative and hence group embeddable. |
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18h |
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What are the main structure theorems on finitely generated commutative monoids? A finite commutative semigroup has a grading by a semilattice such that the homogeneous components are nilpotent extensions of abelian groups. The buzzword is semilattice of Archimedean semigroups. I think Grillet will give the best results on such decompositions. |
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18h |
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What are the main structure theorems on finitely generated commutative monoids? Another big result is that the first order theory is decidable. I can't recall the reference but Mark Sapir knows it. Also finitely generated commutative monoids are residually finite. A lot more is known form numerical and affine semigroups, eg, subsemigroups of N and of Z^m. |
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1d |
answered | Random walk on the hypercube |
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May 15 |
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Transformation terminology question I am not on cstheory stackexchange. The answer to your question is the directed power graph of the semigroup, or at least groupprops.subwiki.org/wiki/… says this is the name in group theory for this. I think the undirected version is more commonly studied, see this paper sciencedirect.com/science/article/pii/… by Peter Cameron et al. |
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May 15 |
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Transformation terminology question @Chad, it seems then that it is the cycle/stem in the semigroup you are interested in. The situation I refer to in 2 is s=[1,2,1,2]. Then on the level on {0,1,2,3} one has s cycles {1,2} and has two branches 0->1 and 2<-3. But on the semigroup it is still index 2, period 2. |
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May 15 |
accepted | Transformation terminology question |
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May 15 |
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Grothendieck fibrations and classifying spaces I'm totally not an expert on this, so I may be saying nonsense but doesn't one have by a result of Thomason that BD is homotopy equivalent to a homotopy colinit of the classify spaces of these groupoids induced by the action of C, or something like that? |
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May 15 |
answered | Transformation terminology question |
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May 5 |
accepted | Unbounded metrics on groups |
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May 4 |
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Unbounded metrics on groups Thanks Yves. I realized only the intro talks about this but I didn't have a link to Bergman's paper. |
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May 4 |
answered | Unbounded metrics on groups |
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May 4 |
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Actions of Thompson group F arxiv.org/abs/1105.4017 might be relevant. |
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May 1 |
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Friedman and proof of Hanna Neumann Conjecture Friedman's paper has been accepted to Memoirs ams.org/cgi-bin/mstrack/accepted_papers/memo and also has a simplification by Dicks. |
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Apr 30 |
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Resolutions chain homotopic to projective ones Can you say any more if the chain homotopy from the projective resolution to the other resolution is surjective at each chain module? |
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Apr 29 |
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Resolutions chain homotopic to projective ones added 10 characters in body |
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Apr 29 |
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Resolutions chain homotopic to projective ones I mean chain homtopy equivalent. I realize they do the same job from the point of view of computing derived functors but I would still like to know if the complex is projective. |
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Apr 29 |
asked | Resolutions chain homotopic to projective ones |
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Apr 27 |
answered | Fixed point theorems |
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Apr 25 |
revised |
Algebras with finite essential arity added 5 characters in body |
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Apr 25 |
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Algebras with finite essential arity By the way the free semigroup belongs to the variety generated by all finite semigroups of finite essential arity so you cannot in some sense describe this property by satisfiability of usual identities. But locally trivial semigroups form a pseudovariety and can be defined by a single pseudoidentity. |
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Apr 25 |
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Algebras with finite essential arity I gave now a self-contained proof that the three-element nilpotent cyclic semigroup has essential arity 3 but is not strongly abelian. Every 2-element semigroup of finite essential arity is strongly abelian so this example is minimal. |
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Apr 25 |
revised |
Algebras with finite essential arity added 346 characters in body |
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Apr 24 |
revised |
Algebras with finite essential arity added 606 characters in body; added 254 characters in body; edited body |
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Apr 24 |
answered | Algebras with finite essential arity |
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Apr 20 |
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Associative algebras with Jacobson radical of codimension 1 Some people would say split basic and use basic for the radical quotient being a direct product of division rings. |
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Apr 19 |
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Why do mathematicians prefer one definition over the other when they both define the same concept? Post category theory it has become clear that it is often cleaner to define an object (up to canonical iso) by a universal property rather than explicit construction. This helps to clarify where it stands in relation to other objects. |
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Apr 16 |
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Examples of common false beliefs in mathematics. added 81 characters in body |
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Apr 16 |
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Examples of common false beliefs in mathematics. I was going to rollback because it would seem the edits should be comments (especially as they are not by the OP, even if this is a CW). I think the edits should probably be comments. |
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Apr 16 |
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Cohomological dimension of groups & number of generators Since the question has been answered in comments I have voted to close as no longer relevant. |
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Apr 15 |
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Can you prove that Average(f(x)) is not equal to f(average(x)) for non-linear f in more than one variable I think mathstackexchange is the appropriate forum for your question. This one is for research-oriented math. Please try there and good luck. |
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Apr 13 |
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Quivers for algebras which are not basic or unital. For unital non-basic algebras there is a unique up to isomorphism unital basic algebra which is Morita equivalent to it and one uses that algebras quivers. I don't have a good answer for the nonunital case unless you want to add a unit. |
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Apr 12 |
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Character table of Sn Should be CW it would seem. |
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Apr 11 |
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Simplicial chain complex with ordered simplices You can also find this in Munkres book. |
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Apr 10 |
asked | Integral Leray Number? |
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Apr 10 |
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When is Ad(pi) an irreducible representation ? If $\pi$ is irreducible, then it is not difficult to show that $M^n(\mathbb C)$ is an irreducible $G\times G$-module via $(g,h)A = \pi(g)A\pi(h)^{-1}$. Now the $G$-module structure you are considering is the restriction of this action to the diagonal $\Delta(G)$. So basically, this can be thought of as a Frobenius reciprocity type of question. |
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Apr 8 |
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Reference for ultrametric spaces Automata, dynamical systems and infinite groups, with V.V.Nekrashevich, V.I.Sushchanskii, Proc. Steklov Inst. Math. v.231 (2000), 134-214 gives the description of ultrametric spaces in terms of trees. |
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Apr 8 |
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IBN for algebraic theories His argument is for varieties of algebras with finitary operations. I don't believe that compact spaces form a variety in this sense. I think that it cannot be defined with just finitary operations. For instance, one has operations of taking limits along ultrafilters built in I believe. Of course E does cover boolean algebras. |
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Apr 8 |
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Self-containing structures There are uncountable many isomorphism classes of 2-generated groups. |
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Apr 7 |
accepted | A flag complex is contractible iff the underlying graph is…? |
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Apr 7 |
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IBN for algebraic theories Also note that if a variety has a non-trivial finite algebra then it has IBN for finite sets by counting homomorphisms to this object. |
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Apr 7 |
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IBN for algebraic theories added 74 characters in body; added 10 characters in body |
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Apr 7 |
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Existence of unknowable algorithms ? +1............. |
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Apr 7 |
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Existence of unknowable algorithms ? Perhaps what he wants is an example of a problem with both a uniform version and a non-uniform version where the non-uniform version is solvable in every instance but the uniform problem is undecidable. This is what Peter Shor does. See also my old question mathoverflow.net/questions/72197/… |
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Apr 7 |
answered | IBN for algebraic theories |
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Apr 6 |
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Existence of unknowable algorithms ? The rough answer that all the answers have in common is that if some piece of information is guaranteed to be finite then there is a Turing machine that has this information preprocessed and can do anything algorithmic with this information. But if you don't know explicitly this information then you will not be able to explicitly write down the Turing Machine. |
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Apr 4 |
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Embedding a semigroup into a divisible semigroup added 38 characters in body |
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Apr 4 |
answered | Embedding a semigroup into a divisible semigroup |
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Apr 1 |
revised |
Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? added 268 characters in body |
