\(D\)-set and groups (Q1907583)

The author introduces the following two definitions. A nonempty set \(L\) with partial binary operation \(\ominus\) is called a difference set if the following conditions are true: (d1) For any \(a\in L\Rightarrow a\ominus a\in L\), (d2) If \(a,b,a\ominus b\in L\), then \(a\ominus (a\ominus b)=b\), (d3) If \(a,b,c, a\ominus b\), \(b\ominus c\in L\), then \(a\ominus c\in L\) and \((a\ominus c)\ominus (a\ominus b)= b\ominus c\). A difference set \(L\) is called a group difference set if in addition to conditions (d1)--(d3) the following condition is true: (d4) \(a\ominus b\in L\) if and only if \(b\ominus a\in L\). Some properties of (group) difference sets are investigated.