TITLE: Conjugation Quandles and... "Quandle-Groups"? From quandles to Groups
QUESTION [10 upvotes]: This question is already asked MathSE
A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure. 

1) $a*a=a$
2) $(a*b)*c=(a*c)*(b*c)$
3) $(a*b) /b=(a/b)*b=a$

If we have a group $(G, \cdot, e,^{-1})$ and $*$ is the cojugation operation on $G$
$$a*b:=bab^{-1}$$
and 
$$a/b:=a*b^{-1}$$
then $(G,*,/)$ is denoted with $Conj(G)$ and is a quandle because it satisfies the quandles axioms

1) $a*a=a=aaa^{-1}$
2) $(a*b)*c=(a*c)*(b*c)$ 

because $c(bab^{-1})c^{-1}=(cbc^{-1})cac^{-1}(cb^{-1}c^{-1})=cbab^{-1}c^{-1}$

3) $(a*b) *b^{-1}=(a*b^{-1})*b=a$

because $b^{-1}(bab^{-1})b=b(b^{-1}ab)b^{-1}=a$
I read that we have too that a group homomorphism between two groups $G$ and $G'$ is a quandle homomorphism between theire cojugation quandles $Conj(G)$ and $Conj(G')$ and that makes $Conj$ a functor betwen the category of Groups and the category of quandles...

I wanted to know more about this functor $Conj$ that "maps" Groups to Quandles and since I'm not expert of category theory I apologize if I use a wrong terminology

$Q1a$ - I learnt that not every Quandle is a conjugation Quandle or in
  other words $conj$ is not ""surjective"" on the "set" of all quandles
  (that is not a set but a class i think) so how can I prove that a
  Quandle is a Conjugation Quandle too? 
$Q1b$ -Wich extra "axioms" must hold in a Quandle that is a
  Conjugation Quandle?

This has something to do with the inverse construction of $Conj$ so my next question is

$Q2$ - There is a way to define a group operation starting with a
  quandle operation? Like an inverse $Conj$ construction that build a "Quandle-Group" $Conj^{-1}(Q)$ from a conjugation quandle $Q$.
$Q3$ Is this process not unique?

With not unique I mean that is possible to have two different groups $G=(G,\cdot,\phi)$ and $G'=(G,\circ,\varphi)$ and $$Conj (G)= Conj(G')$$
this should mean that is possible to have 
$$b\cdot a\cdot \phi(b)=b \circ a \circ \varphi(b)$$
where $\phi(b)$and $\varphi(b)$ are the inverse functions and
$$a \cdot b \neq a \circ b$$

$Q4$ My last question is if possible to generalize the conjugation
  operation of groups for monoids and semigruops in a way that these
  "Monoid-conjugations" and "Semigroup-conjugations" are quandles. If is possible I would like to read more about.

REPLY [5 votes]: The notion of a semigroup with conjugation can be formalized by the notions of an LD-monoid, LRD-monoid, and LRDQ-monoid. The information in this answer can be found in the book Braids and Self-Distributivity by Patrick Dehornoy.
$\textbf{LD-monoids}$
An LD-monoid (LD stands for left-distributive) is an algebra $(M,\cdot,1,\wedge )$ such that $(M,\cdot,1)$ is a monoid and the following identities are satisfied:
$$x\cdot y=(x\wedge y)\cdot x,(x\cdot y)\wedge z=x\wedge (y\wedge z),
x\wedge (y\cdot z)=(x\wedge y)\cdot(x\wedge z),1\wedge x=x,x\wedge 1=1.$$
If $(M,\cdot,1,\wedge )$ is an LD-monoid, then $(M,\wedge )$ is generally not the quandle. However, $(M,\wedge )$ still satisfies the left-distributivity law
$x\wedge (y\wedge z)=(x\wedge y)\wedge (x\wedge z)$. Algebras that satisfy the law
$x\wedge (y\wedge z)=(x\wedge y)\wedge (x\wedge z)$ are commonly called LD-systems and LD-systems are still of interest to at least some knot theorists even though they are usually not quandles..
The operation $\wedge$ acts as a sort of conjugation operation on monoids.
For instance, suppose that $(M,\cdot,1)$ is a monoid where for all $x,y\in M$ there exists a unique $z$ where $z\cdot x=x\cdot y$ (in particular, if $M$ is a group). Then $(M,\cdot,1)$ becomes an LD-monoid when we attach the operation $x\wedge y$ defined by $(x\wedge y)\cdot x=x\cdot y.$ Furthermore, if $(G,\cdot,1,\wedge)$ is an LD-monoid and $G$ is a group, then $x\wedge y=x\cdot y\cdot x^{-1}$.
One can even construct LD-monoids by taking a collection of functions and a sort of conjugacy operation. For example, let $\mathfrak{I}_{\infty}$ be the collection of all injective functions from $\mathbb{N}$ to $\mathbb{N}$. Then $(\mathfrak{I}_{\infty},\cdot,1,\wedge)$ is an LD-monoid where $\cdot$ is the composition function, $1$ is the identity mapping, and $(f\wedge g)(x)=f\circ g\circ f^{-1}(x)$ whenever $x\in Im(f)$ and where $(f\wedge g)(x)=x$ otherwise. 
$\textbf{LRDQ-monoids and quandles}$
Even though LD-monoids do not give us quandles, one can generalize the notion of an LD-monoid to LRDQ-monoids and LRDQ-monoids do give us quandles.
An LRD-monoid is an algebra $(M,\cdot,1,\wedge,\vee)$ that satisfies the following identities. 
$$1\wedge x=1\vee x=x,x\wedge 1=x\vee 1=1;x\cdot y=(x\wedge y)\cdot x=y\cdot(y\vee x)$$
$$(x\cdot y)\wedge z=x\wedge(y\wedge z),(x\cdot y)\vee z=y\vee(x\vee z)$$
$$x\wedge(y\cdot z)=(x\wedge y)\cdot(x\wedge z),x\vee(y\cdot z)=(x\vee y)\cdot(x\vee z).$$
An LRDQ-monoid is an LRD-monoid that satisfies the identity $x\wedge(x\vee y)=x\vee(x\wedge y)=x$. Every group becomes an LRDQ-monoid with operations $\wedge$ and $\vee$ defined by $x\wedge y=xyx^{-1},x\vee y=x^{-1}yx$.
If $(M,\cdot,1,\wedge,\vee)$ is an LRDQ-monoid, then $\wedge$ is idempotent if and only if $\vee$ is idempotent. If $(M,\cdot,1,\wedge,\vee)$ is an LRDQ-monoid such that $\wedge$ and $\vee$ are idempotent, then $(X,\wedge,\vee)$ is a quandle.
For example, suppose that $(M,\cdot,1)$ is a monoid such that for all $x,y$ there are
unique $z_{L},z_{R}$ such that $xy=z_{L}x=xz_{R}$. Then one may define unique operations $\wedge$ and $\vee$ on $(M,\cdot,1)$ such that $x\cdot y=(x\wedge y)\cdot x$ and $x\cdot y=y\cdot(y\vee x)$. Then $(M,\cdot,1,\vee,\wedge)$ is an LRDQ-monoid, and the operations $\vee$ and $\wedge$ are idempotent, so $(M,\vee,\wedge)$ is a quandle.
LD-monoids originally to axiomatize the identities satisfied by the algebra of elementary embeddings obtained from the I3 axiom which states that there exists some non-trivial elementary embedding $V_{\lambda}\rightarrow V_{\lambda}$. The collection of elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$ becomes an LD-monoid. The I3 axiom is one of the strongest large cardinal axioms that far extends the axioms of set theory.