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While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over something.

I am currently thinking of

(1) Linear independence in linear algebra, for elements of a $R$-module,

(2) Algebraic independence for elements of a $K$-algebra (over a field $K$), on which there is a natural action of $K[x_1,\dots,x_n]$,

(3) Free groups, presentation by generators and relations.

(4) Independence in probability.

(5) Shelah's notion of abstract independence for types in model theory, particularly in stable theories.

Q1. Any other natural/interesting examples ? Or comments about these ones ?

Whereas I clearly see a connexion between the first 3 items : the question of the existence of a non trivial 'equation' of a certain kind bounding the elements, item (4) seems of a different nature (at least I do not understand its nature).

Q2. Are there analogies or more between these items ? Is there someting that the probabilistic notion of independence shares with say item (1) ? What is a good notion of independance and why ? Why is it usefull ?

These questions are very naïve, so naïve answers are allowed !

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    $\begingroup$ Matroid theory was developed by Whitney as an abstraction of the notion of linear independence. See jstor.org/stable/2371182. $\endgroup$ – Sam Hopkins Mar 13 '16 at 21:29
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    $\begingroup$ We may want to make this question community wiki. $\endgroup$ – Joseph Van Name Mar 14 '16 at 15:30
  • $\begingroup$ @JosephVanName What does it mean/imply to make this question community wiki ? $\endgroup$ – Drike Mar 14 '16 at 21:26
  • $\begingroup$ Drike. Community wiki is for questions in which there is no best answer because there are possibly several different good answers. For instance, I answered about independence in Boolean algebras, but someone else could talk about independence in Matroid theory while another person could talk about a certain model theoretic independence. $\endgroup$ – Joseph Van Name Mar 14 '16 at 22:54
  • $\begingroup$ Perhaps you might be interested in sciencedirect.com/science/article/pii/S0168007216300276 (see Def 3.1) a free version available from math.cmu.edu/~rami/canon-forking_03_03_2016.pdf $\endgroup$ – Rami Grossberg Jul 20 '16 at 16:01
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The notions of algebraic independence (as in linear independence) and probabilistic independence are both captured in the notions of an independent subset of a Boolean algebra and the notion of an independent family of partitions.

Suppose that $B$ is a Boolean algebra. Then a subset $R\subseteq B$ is said to be independent if $R$ freely generates a subalgebra of $B$. Then $R$ is independent if and only if whenever $r_{1},...,r_{n}\in R$ are distinct elements and for all $i$ either $s_{i}=r_{i}$ or $s_{i}=r_{i}'$ we have $s_{1}\wedge...\wedge s_{n}\neq 0$. Recall that a subset $p$ of a Boolean algebra $B$ is a partition if $0\not\in p$, $\bigvee p=1$ and where $a\wedge b=0$ whenever $a,b\in p,a\neq b$. A collection of partitions $\mathcal{P}$ of $B$ is said to be independent if whenever $p_{1},...,p_{n}\in\mathcal{P}$ and $a_{1}\in p_{1},...,a_{n}\in p_{n}$ then $a_{1}\wedge...\wedge a_{n}\neq 0$.

This notion of independence in a Boolean algebra can be thought of as both an algebraic independence and a probabilistic independence. For example, if $R_{1},...,R_{n}$ are events in a probability space with $P(R_{i})\in(0,1)$ for $1\leq i\leq n$ and $R_{1},...,R_{n}$ are independent in the probabilistic sense, then $R_{1},...,R_{n}$ are independent in a Boolean algebraic sense.

The main result about independence in Boolean algebras is the result by Balcar and Franek which states that every infinite complete Boolean algebra $B$ has an independent subset $R\subseteq B$ with $|R|\subseteq|B|$ and hence a free subalgebra $A\subseteq B$ with $|A|=|B|$. One corollary of this result is that if $A,B$ are complete Boolean algebras with $|A|\leq|B|$, then there is a surjective Boolean algebra homomorphism $\phi:B\rightarrow A$. Furthermore, Balcar and Franek go on to show that complete Boolean algebras have not just large free algebras, but they also have large independent sets of large partitions in the following sense.

Suppose that $B$ is a Boolean algebra. Recall that the saturation $Sat(B)$ of $B$ is the least cardinal such that every partition of $B$ has cardinality less than $Sat(B)$. Then $B$ is said to be well-semifree if there is an independent family $\mathcal{P}$ of partitions such that if $\lambda<Sat(B)$, then $\{p\in\mathcal{P}:|p|=\lambda\}=Sat(B)$. A Boolean algebra $B$ is said to be saturation homogeneous if whenever $B\simeq A\times C$ and $|A|>1$ then $Sat(A)=Sat(B)$.

$\mathbf{Theorem}$(Balcar and Franek) Every saturation homogeneous complete Boolean algebra is well-semifree.

The condition that the complete Boolean algebra $B$ is saturation homogeneous a very minor restriction since every complete Boolean algebra is a direct product of saturation homogeneous complete Boolean algebras.

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Perhaps you might be interested in Definition 3.1 in sciencedirect.com/science/article/pii/S0168007216300276 (link); a free version is available from here.

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