Sunday, August 28, 2016

Algebraic characteristic for sets of infinite measure (Antiideal).

It's a well known fact that every $\sigma$-algebra $\mathcal{F}$ of measurable subsets of $\Omega$ can be thought as a boolean algebra with correspondence

$$ \emptyset \cong 0 \quad  \Omega \cong 1  \quad A \cap B \cong A \wedge B  \cong AB
\quad A \triangle B \cong A \oplus B \cong A + B  .

It is also a well known fact that with this notation $(\mathcal{F},\cdot,+)$ is a commutative ring and for any meaure $\mu$ set $I_0 = \{ A \in \mathcal{F} : \mu(A) = 0 \}$ is a  $\sigma$-ideal of this ring.

Now let $\mu$ be an infinite measure over measurable space $(\Omega,\mathcal{F})$ (assume $\mu(\emptyset)=0$) and define $A_\infty =\{ A \in \mathcal{F} : \mu(A) = \infty \}$. At the first glance it seems that this set doesn't have any intersting algebraic structure. However, now I feel that there is some parallelism between structure of $A_{\infty}$ and group of units of arbitrary nontrivial commutative ring $R$, denote it by $U(R)$ and denote nilradical of $R$ by $I_0(R)$ . Then

$$ 1 \in A_\infty \quad 1 \in U(R) $$
$$ 0  \not \in A_\infty \quad 0 \not \in U(R) $$
$$ \forall b \in A_{\infty}^\complement \forall a \in A_{\infty} \; a +b \in A_\infty
 \forall b \in I_0(R) \forall a \in U(R) \; a +b \in U(R)
$$ \forall b \in A_{\infty}^\complement \forall a \in A_{\infty} \; ab   \not \in A_\infty
 \forall b \in U(R)^\complement \forall a \in U(R) \; ab \not\in U(R)
Clearly $A_\infty \neq U(\mathcal{F}) = \{ \Omega \}$, however we still can see some common structure on this sets, so there must be some name for it (I would call such sets antiideals, but I don't think this is correct at all).


Now I googled actual definition of antiideal at nLab.
That is set $A \subset R$ is an antiideal of $R$ if

 1. $0 \not \in A$

 2. if $a + b \in A$ then $a \in A$ or $b \in A$

 3. if $ab \in A$ then $a,b \in A$

Clearly we can see that if by definition $0 \not \in A_{\infty}$ by definition of $A_{\infty}$ (1. holds).

$\infty =\mu(a \triangle b) \le \mu(a \cup b) \le \mu(a) + \mu(b)$ so either $a \in A_\infty$ or $b \in A_\infty$ (2. holds).

$\infty = \mu(a \cap b) \le \mu(a)$ so $a \in A_\infty$ (3. holds).

This shows that $A_\infty$ is indeed an antiideal of $\mathcal{F}$.

Clearly $U(R)$ is not generaly an antiideal. Consider ring $\mathbb{Z}$ with $U(\mathbb{Z}) = \{ 1, -1 \}$ . Then $(3 - 2) \in U(\mathbb{Z})$, however neither $3$ nor $-2$ is a unit.

So the question is still open.

Friday, February 19, 2016

Exponential Law in Topology

Exponential Law in Topology

Exponential law is well known in the context of classical algebra, namely $(a^b)^c = a^{bc}$.
In  the set-theoretic context maps between from a set $X$ to a set $Y$ are often denoted by $Y^X$. Is this a coincident? It turns out  that not only that  cardinality of corresponding sets can be expressed by power relation $\left|Y^X \right| = |X|^{|Y|}$ , but it also has a certain form of exponential law: $(Z^X)^Y \cong  Z^{X \times Y} $  or in type theoretic $a$ notation (which I personally find way more convenient) $Y \to X \to Z \cong  X * Y \to Z$.

 To prove our last statement it is sufficient to mark out natural bijection $I$:$$ \begin{array} I I & :: & (Y \to X \to Z) \to (X * Y\to Z) \\ I(f)(x,y) &=& f(y)(x) \\I^{-1}(g)(y)(x) &=& g(x,y) \\ \end{array} $$ However, it must be noted that that this fact is established for now only for $\mathsf{SET}$ Category (have we mentioned any others?). As an uncultured tramp I was stroked by the fact that in the category $\mathsf{TOP}$  if we will view $X \to Z$ as space of continuous functions endowed with compact-open topology exponential law holds only then  $X$ is locally compact and Hausdorff (or just compact if you are an algebraic geometrician) and $Y$ just Hausdorff (this conditions turns out to be sufficient. but not necessary) . At first it seemed unbelievable that natural bijection is able to distort continuity properties of maps.   Actually this made my barbaric uncultured blood so hot that that I spent several pointless hours trying to disprove this result.

As it turned out letter, this problem also bothered many topologists in the past.  For example, Edwin Spanier proposed concept of quasitopology  for overcoming this issue.  

Edwin Spainer (1921 -1996)

 Problem of exponential law in topological categories found its development in many other works also. The modern state of the subject can be checked out at this nLab article.

The morale of the story is that this was a test blog entry. And possibly there will be more.