Co-theory of sorted profinite groups for PAC structures

Hoffmann, Daniel Max; Lee, Junguk

doi:10.1142/s0219061322500301

Cited by 2 publications

(11 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The functions J * ⊆ and J * ∩ are necessary to axiomatize the sorted complete system of a sorted profinite group in an appropriate first order language so that the sorted complete system satisfies the modular lattice axiom (cf. [10,Definition 3.7]). Definition 2.2.…”

Section: Sorted Profinite Groupmentioning

confidence: 99%

“…A typical example of a sorted profinite group comes from the Galois group of a first order structure by attaching additional data to each open normal subgroup of the Galois group, which recognize on which sorts a primitive element of a finite Galois extension of the structure live (cf. [10,Section 2]). So, "sorted" in a sorted profinite group means that each open normal subgroup is sorted in a certain way.…”

Section: Sorted Profinite Groupmentioning

confidence: 99%

“…Note that there is a finite tuple a ∈ acl(K ) N such that acl(K ) N = dcl(a, K ) and we call such an element a primitive element of acl(K ) N over K (cf. [10,Fact 2.8]). An extension…”

Section: Sorted Profinite Groupmentioning

confidence: 99%

“…Namely, for each N ∈ N (G(K )) and for a finite tuple a in acl(K ) N , dcl(a, K ) = acl(K ) N ⇔ dcl(a, K ) = acl(K ) res -1 [N ] so that F (N ) ⊆ F (res -1 [N ]) (cf. [10,Fact 2.10]).…”

Section: The Category Of Sorted Profinite Groupsmentioning

confidence: 99%

“…In this article, we consider the embedding property in the category of sorted profinite groups. In [10], we introduced a notion of sorted profinite groups to study the (Shelah-)Galois groups of first order structures. The Galois groups of first order structures are typical examples of sorted profinite groups (see Example 2.5).…”

mentioning

confidence: 99%

See 4 more Smart Citations

The Embedding Property for Sorted Profinite Groups

Lee

2023

J. symb. log.

View full text Add to dashboard Cite

We study the embedding property in the category of sorted profinite groups. We introduce a notion of the sorted embedding property (SEP), analogous to the embedding property for profinite groups. We show that any sorted profinite group has a universal SEP-cover. Our proof gives an alternative proof for the existence of a universal embedding cover of a profinite group. Also our proof works for any full subcategory of the sorted profinite groups, which is closed under taking finite quotients, fibre products, and inverse limits. We also show that any sorted profinite group having SEP has a sorted complete system whose theory is $\omega $ -categorical and $\omega $ -stable under the assumption that the set of sorts is countable.

show abstract

Section: Sorted Profinite Groupmentioning

confidence: 99%

Section: Sorted Profinite Groupmentioning

confidence: 99%

“…Note that there is a finite tuple a ∈ acl(K ) N such that acl(K ) N = dcl(a, K ) and we call such an element a primitive element of acl(K ) N over K (cf. [10,Fact 2.8]). An extension…”

Section: Sorted Profinite Groupmentioning

confidence: 99%

Section: The Category Of Sorted Profinite Groupsmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

The Embedding Property for Sorted Profinite Groups

Lee

2023

J. symb. log.

View full text Add to dashboard Cite

show abstract

On Rank Not Only in Nsop Theories

DOBROWOLSKI,

HOFFMANN

2024

J. symb. log.

Self Cite

View full text Add to dashboard Cite

We introduce a family of local ranks $D_Q$ depending on a finite set Q of pairs of the form $(\varphi (x,y),q(y)),$ where $\varphi (x,y)$ is a formula and $q(y)$ is a global type. We prove that in any NSOP $_1$ theory these ranks satisfy some desirable properties; in particular, $D_Q(x=x)<\omega $ for any finite tuple of variables x and any Q, if $q\supseteq p$ is a Kim-forking extension of types, then $D_Q(q)<D_Q(p)$ for some Q, and if $q\supseteq p$ is a Kim-non-forking extension, then $D_Q(q)=D_Q(p)$ for every Q that involves only invariant types whose Morley powers are -stationary. We give natural examples of families of invariant types satisfying this property in some NSOP $_1$ theories. We also answer a question of Granger about equivalence of dividing and dividing finitely in the theory $T_\infty $ of vector spaces with a generic bilinear form. We conclude that forking equals dividing in $T_\infty $ , strengthening an earlier observation that $T_\infty $ satisfies the existence axiom for forking independence. Finally, we slightly modify our definitions and go beyond NSOP $_1$ to find out that our local ranks are bounded by the well-known ranks: the inp-rank (burden), and hence, in particular, by the dp-rank. Therefore, our local ranks are finite provided that the dp-rank is finite, for example, if T is dp-minimal. Hence, our notion of rank identifies a non-trivial class of theories containing all NSOP $_1$ and NTP $_2$ theories.

show abstract

Co-theory of sorted profinite groups for PAC structures

Cited by 2 publications

References 25 publications

The Embedding Property for Sorted Profinite Groups

The Embedding Property for Sorted Profinite Groups

On Rank Not Only in Nsop Theories

Contact Info

Product

Resources

About