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Let $F$ be a set and $.$ be a binary operation on $F$ and $.^{-1}:F\to F$ be a so-called inverse operation on $F$ such that $(F,.)$ is semigroup and for each $x,y\in F$,

$$(x^{-1})^{-1}=x,~~~~~ (xy)^{-1}=y^{-1}x^{-1}$$

Then $(F,.,^{-1})$ is called an Ef Stucture.

An ef structure is called derived if it is isomorphic to some $(\mathcal F,.,^{-1})$ where $\mathcal F\subseteq \mathcal P(G)$ for a group $G$.

Which ef structures are derived? Is there an algebraic (structural) characterization?

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  • $\begingroup$ Do you also require $(x.y)^{-1}=(y^{-1}).(x^{-1})$? $\endgroup$
    – UwF
    Jun 22, 2014 at 11:25
  • $\begingroup$ @UwF: Clearly it is a requirement to be derived. Yes It can be added to the definition of ef structure. $\endgroup$
    – H. Khas
    Jun 22, 2014 at 11:46
  • $\begingroup$ I think you are suggesting that $P(G)$ carries such a structure. Are you really asking which Ef structures occur as substructures of such $P(G)$? or just asking for abstract characterizations of $P(G)$ (which is what the title suggests)? $\endgroup$
    – Todd Trimble
    Jun 22, 2014 at 12:20
  • $\begingroup$ As substructure. $\mathcal F\subseteq \mathcal P(G)$ is a (derived) ef structure iff it is closed under multiplication and inverse. $\endgroup$
    – H. Khas
    Jun 22, 2014 at 12:23
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    $\begingroup$ If you have for all $x, y$ that $x . x^{-1} = y . y^{-1}$, then you have an associative binary operation with identity and right inverses (by the group definition), which is enough to give you the group structure. So I think that that's necessary and sufficient. $\endgroup$
    – user44191
    Jul 6, 2014 at 0:05

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