Ekvivalentné definície P-ideálu

[Martin Sleziak]

Niekedy sme spomínali, že existujú viaceré ekvivalentné definície P-ideálu.

• For every sequence $(A_n)_{n\in\mathbb N}$ of sets from $\mathcal I$ there is $A\in\mathcal I$ such that $A_n\subseteq^* A$ for all $n$'s.
• Any sequence $(F_n)_{n\in\mathbb N}$ of sets from $\mathcal F(\mathcal I)$ has a pseudointersection in $\mathcal F(\mathcal I)$. (Pseudointersection of $(F_n)_{n\in\mathbb N}$ is a set $F$ such that $F\subseteq^* F_n$ for each $n\in\mathbb N$.)
• For every sequence $(A_n)_{n\in\mathbb N}$ of sets belonging to $\mathcal I$ there exists a sequence $(B_n)_{n\in\mathbb N}$ of sets from $\mathcal I$ such that $A_j =^* B_j$ for $j\in\mathbb N$ and $B=\bigcup_{j\in\mathbb N} B_j\in\mathcal I$.
• For every sequence of mutually disjoint sets $(A_n)_{n\in\mathbb N}$ belonging to $\mathcal I$ there exists a sequence $(B_n)_{n\in\mathbb N}$ of sets belonging to $\mathcal I$ such that $A_j =^* B_j$ for $j\in\mathbb N$ and $B=\bigcup_{j\in\mathbb N} B_j\in\mathcal I$.
• For every non-decreasing sequence $A_1\subseteq A_2 \subseteq \dots \subseteq A_n \subseteq \dots$ of sets from $\mathcal I$ there exists a sequence $(B_n)_{n\in\mathbb N}$ of sets belonging to $\mathcal I$ such that $A_j =^* B_j$ for $j\in\mathbb N$ and $B=\bigcup_{j\in\mathbb N} B_j\in\mathcal I$.
• In the Boolean algebra $\mathcal P(S)/\mathrm{Fin}$ the ideal $\mathcal I$ corresponds to a $\sigma$-directed subset, that is, every countable subset has an upper bound.

Symbol $A\subseteq^* B$ označuje, že množina $A\setminus B$ je konečná. Symbol $A=^*B$ označuje, že symetrický rozdiel $A\mathrel{\triangle}B$ je konečný.

Ako referencie, kde sa dajú nájsť dôkazy, že niektoré z týchto podmienok sú ekvivalentné, môžem spomenúť Proposition 1 v článku Balcerzak, Marek; Dems, Katarzyna; Komisarski, Andrzej Journal of Mathematical Analysis and Applications, Vol. 328, No. 1, p. 715-72, DOI: 10.1016/j.jmaa.2006.05.040. O niečo všeobecnejšie je to sformulované ako Lemma 3.9 v článku Mačaj-Sleziak týkajúcom sa $\mathcal I^{\mathcal K}$-konvergencie: https://projecteuclid.org/euclid.rae/1300108092

Okrem názvu P-ideál sa bežne vyskytujú aj vlastnosť AP alebo vlastnosť AP0.

[Martin Sleziak]

Ešte v súvislosti s terminológiou spomeniem, že názov P-ideál pravdepodobne pojmu P-bodu (P-point) v topológii. Existuje do istej miery príbuzný pojem P-priestoru (P-space).

Toto je citát z článku A. Blassa:

9.24 Remark.
There is a general topological concept of P-point (see for example [95, 50]), namely a point (in a topological space) such that every countable intersection of open neighborhoods of it includes another open neighborhood of it. When applied to the topological space βω − ω, the Stone-Čech remainder of the discrete space ω, whose points are naturally identified with (non-trivial) ultrafilters on ω, this topological notion becomes the concept defined above.
The “P” in “P-point” refers to prime ideals (in rings of functions); see [50, Exercises 4J and 4L].
The “Q” in “Q-point” was chosen because it is next to “P” in the alphabet. Q-points are also called rare ultrafilters.
...
[50] Leonard Gillman and Meyer Jerison. Rings of Continuous Functions. Van Nostrand, Princeton, 1960.
[95] Walter Rudin. Homogeneity problems in the theory of Čech compactifications. Duke Mathematical Journal, 23:409–419, 1956. (projecteuclid, DOI: 10.1007/978-3-0348-7524-0_7)

Andreas Blass: Combinatorial Cardinal Characteristics of the Continuum (Handbook of Set Theory, Editors: Matthew Foreman, Akihiro Kanamori, 2010, pp 395-489)
http://books.google.com/books?id=DLCyeh ... ology+name
http://dx.doi.org/10.1007/978-1-4020-5764-9_7
http://www.math.lsa.umich.edu/~ablass/hbk.pdf

[Martin Sleziak]

Od Slava Mišíka som sa v maile dozvedel ďalšiu ekvivalentnú charakterizáciu:

Dalsia ekvivalentna definicia je v spominanom clanku Grekos. Misik, Ziman. Hovori, ze pre kazdy I-rozklad N existuje mnozina z dualneho filtra, ktora ma konecny prienik s kazdou mnozinou v rozklade.

Toto sa do istej miery podobá na túto vec, ktorú som však videl iba pre ultrafiltre (t.j. pre maximálne ideály). Bolo by možno zaujímavé vedieť, či niečo podobné platí pre ľubovoľné filtre (nielen ultrafiltre) a či to je niekde spravené.

Zacitujem z knihy Halbeisen: Combinatorial Set Theory, s. 221. (Táto kniha je voľne prístupná na autorovej webstránke.)

A free ultrafilter $\mathscr U$ is a P-point if for each partition $\{u_n\subseteq\omega; n\in\omega\}$ of $\omega$ either $u_n \in \mathscr U$ for a (unique) $n\in\omega$, or there exists an $x \in \mathscr U$ such that for each $n\in\omega$, $x\cap u_n$ is finite.

Furthermore, a free ultrafilter $\mathscr U$ is a Q-point if for each partition of $\omega$ info finite pieces $\{I_n\subseteq\omega; n\in\omega\}$ (i.e., for each $n\in\omega$, $I_n$ is finite), there exists an $x\in\mathscr{U}$ such that for each $n\in\omega$, $x\cap I_n$ has at most one element.

Tieto typy ultrafiltrov súvisia aj s nejakými ďalšími typmi ultrafiltrov, ktoré sa študujú, ako sú napríklad ramseyovské ultrafiltre (Ramsey ultrafilters). Niečo sa dá o nich prečítať na Wikipédii (pre istotu pridám linku aj na poslednú revíziu) alebo v spomenutej knihe.