monoids Questions - MathOverflow

Representations of products of groups (and monoids)

2010-07-30T03:55:13Z

I have very little knowledge of representation theory, but the following has come up in my summer undergrad research project (relates to conformal field theory and geometric function theory).

Suppose we have a group $G$ and subgroups $A$ and $B$ such that $A \cap B = {1}$ and for every $g \in G$, there exists $a \in A, b \in B$ such that $g = ab$. Then we can express $G = A \bowtie B$ as a Zappa–Szép product. This of course reduces to the semidirect or direct product in the nice cases.

Then suppose, we have sufficiently nice representations of $A$ on an $F$-vector space V, and $B$ on an $F$-vector space W, then can we find a representation of $G$ which in some sense preserves the representations of $A$ and $B$?

I've been told that the solution for semidirect products uses something called Clifford Theory, but we don't have a semidirect product here.

Our problem involves a monoid, not a group, but the Zappa-Szep product is constructed the same way there.

What is the relation between the number syntactic congruence classes, and the number of Nerode relation classes?

2010-07-22T04:53:17Z

For a monoid $M$ and a subset $S$ of $M$, define the syntactic congruence $\equiv_S$ of $S$ as the least congruence on $M$ that saturates $S$, i.e. : $$u \equiv_S v \Leftrightarrow (\forall x, y)[xuy \in S \leftrightarrow xvy \in S].$$

Now define the Nerode equivalence as the following right congruence : $$u \sim_S v \Leftrightarrow (\forall x)[ux \in S \leftrightarrow vx \in S].$$

Let $[u]_\equiv$ be the equivalence class of $u$ with respect to $\equiv_S$ and $[u]_\sim$ with respect to $\sim_S$.

Now define $i_\equiv (n)$ to be the number of different $[u]_\equiv$ for $u$ of size $n$.

Define $i_\sim(n)$ in a similar fashion.

Now the question is, how do the two $i$ functions relate ?

For instance, a standard theorem says that $i_\sim(n)$ is bounded by a constant whenever $i_\equiv(n)$ is, and reciprocally. Is there any other result in this trend?

Vocabulary on monoid periodicity

2010-07-17T02:53:47Z

I'm learning about periodic languages, and I'm confused over the vocabulary used to describe the periodicity of (syntactic) monoids.

If I understand correctly, a monoid M is periodic if : $$(\forall m \in M)(\exists i \neq j)[m^i = m^j],$$ and it is aperiodic if : $$(\exists k)(\forall m \in M)[m^k = m^{k+1}],$$ and then an aperiodic monoid is periodic. Where does that bizarre vocabulary come from?

And in the same vein, what would be the book you'd recommend on monoids and semigroups?

Thank you.

Associative binary operations on natural numbers

2010-07-09T11:51:31Z

Which are all the associative binary operations on natural numbers ? Certain results in this regard can be found in arxiv:math/0508215. It appears that such associative operations cannot grow too fast. Or perhaps, there are also those which have to grow very fast. It is easy to prove that there are infinitely many such associative operations. And in fact, uncountably many, as shown in the mentioned paper. Amusingly, under rather simple natural conditions, the usual addition and multiplication are the only associative binary operations on natural numbers, as wes proved back in 1981, and cited in the mentioned paper. This fact is the motivation for the above question, namely, what happens if we do not ask any conditions on such associative binary operations ? The whole story arose from the observation that certain minimal algebraic and topological axioms determine uniquely the sets on which they hold. An example is the theorem of Pontriaghin proved in the 1930s that the only commutative and algebraically closed field which is not discrete and it is complete is that of complex numbers. And then, one can turn the issue around and ask whether there are simplest infinite sets which determine uniquely some of the algebraic, topological, or other structures on them ? Well, it turns out the the set of natural integers just about determines uniquely addition and multiplication.

$\omega$-monoids

2010-06-30T18:28:11Z

Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.

This is an attempted rephrasing of question: http://mathoverflow.net/questions/24723/chain-hierarchy-of-monoids. My application domain is reasoning about modifiers of modifiers in software product line engineering, thus lacking established mathematical background. I find it easier to adapt existing results, even if the application domain is significantly different, so any help would be appreciated.

Structure Theorem for finitely generated commutative cancellative monoids?

2010-06-20T08:08:56Z

Is there a Structure Theorem for finitely generated commutative cancellative monoids?

Of course they can be densely embedded into a finitely generated abelian group, whose structure is known. Also, in the book of J. C. Rosales and P. A. García-Sánchez there are some special embedding theorems: If the monoid is torsionfree, it even embeds to some free abelian group, and if the monoid is also reduced, it embeds in some free commutative monoid.

But I want to know if it is possible to give a complete classification (for example, in terms of generators and relations, as in the case of groups).

Chain/Hierarchy of Monoids

2010-05-15T10:45:40Z

Let's assume that we have the following collection of structures:

Some space $P$.
Monoids $(M_{i+1},\circ_{i+1})$, and
Actions $\bullet_{i+1}:M_{i+1}\times M_i\to M_i$, for $i\ge 0$
And $\bullet_{0}:M_0\times P\to P$.

satisfying

($\bullet$ is a monoid action): $(m\circ_{i+1}m')\bullet_{i+1} n = m\bullet_{i+1}(m'\bullet_{i+1} n)$ and
($m\bullet-$ is a homomorphism): $m\bullet_{i+1}(n\circ_{i}n')=(m\bullet_{i+1}n)\circ_{i} (m\bullet_{i+1} n')$.

In my application, $P$ corresponds to computer programs. $M_0$ are modifications to elements of $P$. If you wish, you can think of $M_0$ as some kind of structured patch. Then each $M_{i+1}$ are higher-order modifications of the modifications in $M_i$.

The hierarchy isn't necessarily infinite.

I'm curious to know what kind of structure I'm looking at. I originally felt that I was defining some kind of $n$-category with one object at each level, namely the endomorphism, but one reader commented that my structures were too floppy, meaning that there were not enough equations.

It seems that the structure I'm interested in is related to the automorphism tower for groups, except that I'm interest in monoids, and rather than automorphism, I'm only concerned with endomorphism, and I am working indirectly through monoid actions, rather than having the endomorphism apply to the morphisms at the level below.

Have I defined a known structure?

What natural equations would one expect to link the various levels with each other?

What additional properties does it satisfy? What reasonable properties should it satisfy?

Are there conditions under which it becomes degenerate?

Any pointers would be appreciated.

Torsors for monoids

2010-05-25T10:33:02Z

Torsors are defined as a special kind of group action. I am wondering whether the analogous notion exists for monoid actions. Some references would be helpful.

In general I'm interesting in the notion of 'subtraction/division' induced by having a torsor. My application is in computer science. A monoid is used to capture modifications to a computer program and the monoid action corresponds to performing the modification on the program. If I have a torsor-like entity, I can take two software entities and produce the modification required to convert one into the other.

The answer is that any such monoid will automatically be a group. In my application, it seems that I will only get close to the notion of torsor if my modifications have inverses, which they do not.

Does the category Monoid of monoids have finite coproducts?

2010-03-12T10:18:39Z

Does the category Monoid of monoids have finite coproducts?

commutative monoids have binary products?

2010-03-13T00:39:12Z

Does the category CMonoid of commutative monoids have binary products?

thanks

How do you compute the space of lifts of an E-infinity map?

2010-02-15T21:37:03Z

Let X, Y and B be $E_\infty$ spaces, and let $p: X \rightarrow Y$ and $f: B \rightarrow Y$ be $E_\infty$ maps. We can ask for the space of lifts of f across p, that is the space of $E_\infty$ maps $g: B \rightarrow X$ such that $pg = f$.

Q1: What spectral sequences or other technology exists for computing the (homotopy groups of the) space of such E-infinity lifts?

Any $E_\infty$ lift $g: B \rightarrow X$ provides a lift of the map of commutative monoids $\pi_0 B \rightarrow \pi_0 Y$ across the map $\pi_0 X \rightarrow \pi_0 Y$.

Q2: Do all the techniques for computing E-infinity lifts in effect require that you first solve this $\pi_0$ lifting problem in commutative monoids, and begin an obstruction calculation from there, or are there techniques that solve both the $\pi_0$ problem and the E-infinity problem 'simultaneously' and perhaps in a way that eases both computations?

I am particularly interested in merely knowing if there exists an $E_\infty$ lift, thus might have asked the seemingly more basic question:

Q1': When is there an E-infinity lift of an E-infinity map?

I think the obstruction groups answering Q1' are liable to come packaged in the answers to Q1/Q2, but if there are separate techniques for the existence question, that would also be helpful.

Remark: I could imagine that one answer to Q1 involving relative Andre-Quillen cohomology might be extracted from Goerss-Hopkins, Moduli Problems for Structured Ring Spectra, but perhaps there are more elementary means. I'd be very interested for answers to Q1 along those or especially other lines of thought, and any ideas about Q2. Thanks!

Why is a monoid with closed symmetric monoidal module category commutative?

2010-02-09T21:13:01Z

Given a symmetric monoidal category and a monoid object A in it, one can form the category of modules over this monoid object, i.e. objects are $A \otimes M \rightarrow M$ satisfying the natural properties analogous to modules over a ring and morphisms respecting this. The following seems to be true and I would like to know why:

If the category of modules has a closed symmetric monoidal structure with A as unit object, then A is a commutative monoid.

This is how I read the statement right after Proposition 2.3.4 in Hovey/Shipley/Smith's paper "Symmetric Spectra" and it would give an excellent motivation for introducing symmetric spectra...

monoid ring and some structure within it - how is it called?

2010-02-05T10:45:09Z

I am amateur - mathematics is my hobby, and I find some strange structure working with toy matrices structure so I try to ask some questions regarding it. Let me allow to introduce some structure which I do not understand.

Suppose we have finitely presented monoid with unity $M$ with two generators say $g_1,g_2$. Lets relations for this monoid would be $Rel = \{g_1^2 = e , g_2^2 = g_2 \}$ where $e$ is unit element of monoid. So we have monoid $M$ to be quotient of free monoid $F$ by relations $M = F/ Rel$. $M$ is infinite. Words in $M$ has structure "$stststststst...$" etc. Rather boring ;-)

Now I want to define ring $G$ over such monoid. Lets play with field of real numbers R, as a background field. So we have ring $G = R[M]$. Suppose we are able to find such element, let's call it $g_3$ in R[M] that the following equations are satisfied:

(1) $g_i g_j = c_{ijk}g_k$ where $i,j,k=1,2,3$ just like in Lie algebra structure.

Note that $g_3$ is not element of monoid $M$ but is element of ring $G$. Also there is no antisymmetry relations for $c_{ijk}$. Then note, that from (1) we have that every element in $R[M]$ is linear combination of set of "generators" $\{ e,g_i \} , i=1,2,3$.

In one sentence within ring $R[M]$ we have some structure which allows us to easily compute every polynomial formula as it after some evaluations may be always turned into linear combination of generators. But such generators of $R[M]$ are different that generators of base monoid $M$.

Do You know any references where I may find examples of such structures? How they are called? They are examples of what? Are there any computer algebra systems which compute with such structures?

Additional remarks:

@ Darji - "just like Lie algebra" is about formal structure. It reminds me definition of Lie algebra, but of course $c_{ijk}$ is not antisymmetric nor Jacobi identity is satisfied so of course it is not Lie algebra.

@ Darij - Of course in general there is no associativity. In case I am interesting in this structure is associative, as it follow from simple algebra monoid which is associative, and by R[M] I mean formal combinations $\sum a_i g_i$ and combinations of its multiplies as in section "two simple examples" in http://en.wikipedia.org/wiki/Group_ring So we have noncommutative ring over monoid which is associative, has unity, and $c_{ijk}$ in j,k has both symmetric and antisymmetric components.

Further clarifications: Structure I tried to describe consists of multiplicative monoid, and ring over it in reals. In this ring every polynomial has linear decomposition in "basis" $g_i$, somehow as in vector space. In ring every ring element allows such decompositions ( but not every linear combination of $g_i$ is ring element so it is not linear space). What is that? Do You know examples of such structures?

@Scott: You are right I am very bad English writer. Thank You for being so polite. So I will wrote it in the most explicit way I can.

I have finitely presented noncommutative monoid with unity and two generators $g_1,g_2$: M = F/Rel where $Rel = \{g_1^2 = e , g_2^2 = g_2 \}$, $e$ is unit element and $F$ is free monoid over two generators. Because of relations $Rel$ every elemet in monoid has form for example $g = g_1g_2g_1g_2...g_1g_2$ ( alternating finite sequence with subscripts 1212... or 2121...). Different monoid elements contains different number of multiplications. It is very simple although infinite multiplicative structure.

Then I consider monoid ring over reals $R[M]$. Every element in $R[M]$ has form:

(1) $t = r_1g_1 + r_2g_2 r_3g_1g_2+r_4g_2g_1+ r_5g_1g_2g_1 + ...+r_p g_ig_kg_i...g_s+ ...$ and so on. $r_i \in R$ and $g_i \in M$.

Note that in general monomial element $g_ig_k...g_s$ every subscript has value in $\{1,2\}$ and no two following each other subscripts are the same ( they alternate like in sequence like $1212..$ or $2121..$. Of course this is standard ring definition.

In structure, I would like to describe You here, I have strange additional property: there is element $g_3$ in ring $R[M]$ ( but it is not monoid element!) which allows following decomposition:

For every $r \in R[M]$ we have

$r= r_0 e + r_1 g_1 +r_2 g_2 +r_3 g_3$

Look: there are only four terms in decomposition, even if You decompose general ring element in the form of (1). However after such decomposition I may only multiply such elements and not add them. So in fact decomposition as above, I trying to treat as some kind of "parametrization" of ring elements. Is this interesting?

As far as I know this is not standard ring property - maybe I am wrong. If I think about for example polynomial ring (that in simple case is real ring over multiplicative monoid generated by one generator $x$) such decomposition is not possible.

So I ask You if that structure was described in literature? Is it special kind of some known structure? Where to find something about it?

Thank You all for Your remarks!

Homological Algebra for Commutative Monoids?

2009-10-13T16:23:02Z

Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of short exact sequence. Can we use this to define ext groups which classify extensions? What works and what doesn't work and why?

Computing the structure of the group completion of an abelian monoid, how hard can it be?

2010-02-03T07:24:35Z

Cherry Kearton, Bayer-Fluckiger and others have results that say the monoid of isotopy classes of smooth oriented embeddings of $S^n$ in $S^{n+2}$ is not a free commutative monoid provided $n \geq 3$. The monoid structure I'm referring to is the connect sum of knots.

Bayer-Fluckiger has a result in particular that says you can satisfy these equations $$a+b=a+c, \ \ \ \ b \neq c$$ where $a,b,c$ are isotopy classes of knots and $+$ is connect sum.

When $n=1$ it's an old result of Horst Schubert's that the monoid of knots is free commutative on countably-infinite many generators.

What I'm wondering is, does anyone have an idea of how difficult it might be to compute the structure of the group completion of the monoid of knots, say, for $n \geq 3$? That's not really my question for the forum, though.

It's this: Do people have good examples where it's "easy" to compute the group-completion of a commutative monoid, but for which the monoid itself is still rather mysterious? Meaning, one where rather minimal amounts of information are required to compute the group completion? Presumably there are examples where it's painfully difficult to say anything about the group completion? For example, can it be hard to say if there's torsion in the group completion?

Recovering a monoidal category from its category of monoids

2009-10-29T19:34:30Z

What kind of additional properties and/or structures one needs to impose on the category of (commutative or noncommutative) monoids of some monoidal category so that one can recover the original monoidal category from this data?

What kind of additional properties and/or structures one needs to impose on a category to ensure that it is the category of monoids of some monoidal category?

The example I have in mind is the category of (commutative or noncommutative) C*-algebras (or von Neumann algebras). Can we obtain one of these categories as the category of monoids of some monoidal category?