7
votes
1answer
196 views

Über theorem on unavoidable patterns?

Let $A$ be an alphabet of $k$ symbols, and $p$ a pattern. An example of a pattern is $p=XX$, where $X$ is any finite string of symbols from $A^+$. Avoiding $p$ is avoiding any subword repeated twice ...
4
votes
1answer
170 views

Is the monoid of taking iterated images and inverse images freely generated by the image and inverse image operation?

Let $\mathcal{F}$ denote the class of all functions. Let $U,L:\mathcal{F}\rightarrow\mathcal{F}$ denote the mappings where if $f:X\rightarrow Y$, then $U(f):P(X)\rightarrow P(Y),L(f):P(Y)\rightarrow ...
3
votes
0answers
116 views

What is the combinatorial data classifying non-normal affine toric varieties?

Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I ...
4
votes
2answers
268 views

Concatenation of strings [closed]

We have two strings (i. e., finite tuples) $A$ and $B$. We have to find if for some positive integers $n$ and $m$, the string $A$ concatenated $n$ times equals the string $B$ concatenated $m$ times or ...
5
votes
2answers
144 views

Functionals on oriented matroids

Oriented matroids are abstractions of hyperplane arrangements, or equivalently vector configurations. Let me recall the definition in terms of covectors. Let $R=\lbrace 0,+,-\rbrace$ with the monoid ...
8
votes
2answers
215 views

An operation on binary strings

Consider the “product” $\gamma = \alpha \times \beta$ of two binary strings $\alpha$, $\beta$ $\in \lbrace 0,1\rbrace^+$ which one gets by replacing every 1 in $\beta$ by $\alpha$ and each ...
2
votes
2answers
289 views

How many idempotent relations are there on an $n$-element set?

As far as I know, it is an open problem to give a formula counting transitive relations on an $n$-element set. Is it easier to count the idempotent relations, that is relations that are both ...
7
votes
1answer
395 views

Powers of maps on finite sets

Let $X_n$ be a set with $n$ elements. Write $F(X_n,X_n)$ for the set of maps from $X_n$ to itself. It is a monoid under the operation of composition. Let $m$ be a positive integer. How many maps in ...