Maximal WAP and tame quotients of type spaces
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| Formato: | Preprint |
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2025
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| _version_ | 1866913666758082560 |
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| author | Krupiński, Krzysztof Portillo, Adrián |
| author_facet | Krupiński, Krzysztof Portillo, Adrián |
| contents | We study maximal WAP and tame (in the sense of topological dynamics) quotients of $S_X(\mathfrak{C})$, where $\mathfrak{C}$ is a sufficiently saturated (called monster) model of a complete theory $T$, $X$ is a $\emptyset$-type-definable set, and $S_X(\mathfrak{C})$ is the space of complete types over $\mathfrak{C}$ concentrated on $X$. Namely, let $F_{\textrm{WAP}}\subseteq S_X(\mathfrak{C})\times S_X(\mathfrak{C})$ be the finest closed, $aut(\mathfrak{C})$-invariant equivalence relation on $S_X(\mathfrak{C})$ such that the flow $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{WAP}} )$ is WAP, and let $F_{\textrm{Tame}}\subseteq S_X(\mathfrak{C})\times S_X(\mathfrak{C})$ be the finest closed, $aut(\mathfrak{C})$-invariant equivalence relation on $S_X(\mathfrak{C})$ such that the flow $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{Tame}} )$ is tame. We show good behaviour of $F_{\textrm{WAP}}$ and $F_{\textrm{Tame}}$ under changing the monster model $\mathfrak{C}$. Namely, we prove that if $\mathfrak{C}'\succ \mathfrak{C}$ is a bigger monster model, $F'_{\textrm{WAP}}$ and $F'_{\textrm{Tame}}$ are the counterparts of $F_{\textrm{WAP}}$ and $F_{\textrm{Tame}}$ computed for $\mathfrak{C}'$, and $r\colon S_X(\mathfrak{C}')\to S_X(\mathfrak{C})$ is the restriction map, then $r[F'_{\textrm{WAP}}]=F_{\textrm{WAP}}$ and $r[F'_{\textrm{Tame}}]=F_{\textrm{Tame}}$. Using these results, we show that the Ellis (or ideal) groups of $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{WAP}} )$ and $(aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{Tame}})$ do not depend on the choice of the monster model $\mathfrak{C}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15632 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximal WAP and tame quotients of type spaces Krupiński, Krzysztof Portillo, Adrián Logic We study maximal WAP and tame (in the sense of topological dynamics) quotients of $S_X(\mathfrak{C})$, where $\mathfrak{C}$ is a sufficiently saturated (called monster) model of a complete theory $T$, $X$ is a $\emptyset$-type-definable set, and $S_X(\mathfrak{C})$ is the space of complete types over $\mathfrak{C}$ concentrated on $X$. Namely, let $F_{\textrm{WAP}}\subseteq S_X(\mathfrak{C})\times S_X(\mathfrak{C})$ be the finest closed, $aut(\mathfrak{C})$-invariant equivalence relation on $S_X(\mathfrak{C})$ such that the flow $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{WAP}} )$ is WAP, and let $F_{\textrm{Tame}}\subseteq S_X(\mathfrak{C})\times S_X(\mathfrak{C})$ be the finest closed, $aut(\mathfrak{C})$-invariant equivalence relation on $S_X(\mathfrak{C})$ such that the flow $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{Tame}} )$ is tame. We show good behaviour of $F_{\textrm{WAP}}$ and $F_{\textrm{Tame}}$ under changing the monster model $\mathfrak{C}$. Namely, we prove that if $\mathfrak{C}'\succ \mathfrak{C}$ is a bigger monster model, $F'_{\textrm{WAP}}$ and $F'_{\textrm{Tame}}$ are the counterparts of $F_{\textrm{WAP}}$ and $F_{\textrm{Tame}}$ computed for $\mathfrak{C}'$, and $r\colon S_X(\mathfrak{C}')\to S_X(\mathfrak{C})$ is the restriction map, then $r[F'_{\textrm{WAP}}]=F_{\textrm{WAP}}$ and $r[F'_{\textrm{Tame}}]=F_{\textrm{Tame}}$. Using these results, we show that the Ellis (or ideal) groups of $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{WAP}} )$ and $(aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{Tame}})$ do not depend on the choice of the monster model $\mathfrak{C}$. |
| title | Maximal WAP and tame quotients of type spaces |
| topic | Logic |
| url | https://arxiv.org/abs/2501.15632 |