# Benjamin Steinberg

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## Registered User

 Name Benjamin Steinberg Member for 1 year Seen 12 hours ago Website Location New York City Age 40
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.
 18h comment 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. 18h comment 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. 18h comment 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. 18h comment 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. 1d answered Random walk on the hypercube May15 comment Transformation terminology questionI 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. May15 comment 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. May15 accepted Transformation terminology question May15 comment Grothendieck fibrations and classifying spacesI'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? May15 answered Transformation terminology question May5 accepted Unbounded metrics on groups May4 comment Unbounded metrics on groupsThanks Yves. I realized only the intro talks about this but I didn't have a link to Bergman's paper. May4 answered Unbounded metrics on groups May4 comment Actions of Thompson group Farxiv.org/abs/1105.4017 might be relevant. May1 comment Friedman and proof of Hanna Neumann ConjectureFriedman's paper has been accepted to Memoirs ams.org/cgi-bin/mstrack/accepted_papers/memo and also has a simplification by Dicks. Apr30 comment Resolutions chain homotopic to projective onesCan you say any more if the chain homotopy from the projective resolution to the other resolution is surjective at each chain module? Apr29 revised Resolutions chain homotopic to projective onesadded 10 characters in body Apr29 comment Resolutions chain homotopic to projective onesI 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. Apr29 asked Resolutions chain homotopic to projective ones Apr27 answered Fixed point theorems Apr25 revised Algebras with finite essential arityadded 5 characters in body Apr25 comment Algebras with finite essential arityBy 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. Apr25 comment Algebras with finite essential arityI 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. Apr25 revised Algebras with finite essential arityadded 346 characters in body Apr24 revised Algebras with finite essential arityadded 606 characters in body; added 254 characters in body; edited body Apr24 answered Algebras with finite essential arity Apr20 comment 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. Apr19 comment 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. Apr16 revised Examples of common false beliefs in mathematics.added 81 characters in body Apr16 comment 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. Apr16 comment Cohomological dimension of groups & number of generatorsSince the question has been answered in comments I have voted to close as no longer relevant. Apr15 comment Can you prove that Average(f(x)) is not equal to f(average(x)) for non-linear f in more than one variableI think mathstackexchange is the appropriate forum for your question. This one is for research-oriented math. Please try there and good luck. Apr13 comment 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. Apr12 comment Character table of SnShould be CW it would seem. Apr11 comment Simplicial chain complex with ordered simplicesYou can also find this in Munkres book. Apr10 asked Integral Leray Number? Apr10 comment 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. Apr8 comment Reference for ultrametric spacesAutomata, 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. Apr8 comment IBN for algebraic theoriesHis 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. Apr8 comment Self-containing structuresThere are uncountable many isomorphism classes of 2-generated groups. Apr7 accepted A flag complex is contractible iff the underlying graph is…? Apr7 comment IBN for algebraic theoriesAlso note that if a variety has a non-trivial finite algebra then it has IBN for finite sets by counting homomorphisms to this object. Apr7 revised IBN for algebraic theoriesadded 74 characters in body; added 10 characters in body Apr7 comment Existence of unknowable algorithms ?+1............. Apr7 comment 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/… Apr7 answered IBN for algebraic theories Apr6 comment 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. Apr4 revised Embedding a semigroup into a divisible semigroupadded 38 characters in body Apr4 answered Embedding a semigroup into a divisible semigroup Apr1 revised Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?added 268 characters in body