A characterization of Bol left loops. (Q1428275)

Let \((Q,\cdot)\) be a left quasigroup; hence a set \(Q\) together with a map \(\cdot\colon Q\to\text{Sym\,}Q\); \(a\mapsto a^\cdot\). For \(a,b\in Q\) let \(e_a:=(a^\cdot)^{-1}(a)\), \(a^r:=(a^\cdot)^{-1}(e_a)=(a^\cdot)^{-2}(a)\) and \(l_{a,b}:=((a^\cdot(b))^\cdot)^{-1}\circ a^\cdot\circ b^\cdot\). The following properties are considered: (LIP) \(x^\cdot\circ(x^r)^\cdot=\text{id}\) (left inverse property), (LLP) \(l_{a,b}=l_{a^\cdot(b),b}\) (left loop property), (LBI) \(a^\cdot\circ b^\cdot\circ a^\cdot=(a^\cdot(b^\cdot(a)))^\cdot\) (left Bol identity), (R, id) \(\forall a,b\in Q\): \(e_r:=e_a=e_b\) (\((Q,\cdot)\) has a right identity), (L, id) \(\exists e_l\in Q\): \(e_l^\cdot=\text{id}\) (\((Q,\cdot)\) has a left identity). \((Q,\cdot)\) is called `left loop' if (R, id) is satisfied, `quasigroup' if \(Q^\cdot:=\{a^\cdot\mid a\in Q\}\) acts regularly on \(Q\), `loop' if \((Q,\cdot)\) is a quasigroup with (R, id) and (L, id). The authors show: There are quasigroups with (R, id) and (LLP) but where (L, id) and (LBI) are violated; with (LBI) and (R, id) but without (L, id). There are \((Q,\cdot)\) with (\(*\)) \(\exists \text{ permutation }Q\to Q\); \(x\mapsto x^*\): \(x^\cdot\circ(x^*)^\cdot=\text{id}\) but without (R, id). Only if (R, id) holds: ``\((*)\Leftrightarrow\) (LIP)''. ``(LLP) \(\wedge\) (L, id) \(\wedge\) (R, id) \(\Rightarrow\) (LIP) \(\wedge\) (LBI)'', ``(LBI) \(\wedge\) (\(\forall x \in Q\): \((x^r)^\cdot(e_x)=x^r\)) (\ ) \(\Rightarrow\) (LIP) \(\wedge\) (LLP)'', ``(LBI) \(\wedge\) (R, id) \(\Rightarrow\) \(l_{x,y}=(l_{y,x})^{-1}\wedge l_{x,y}=l_{x,y^\cdot(x)}\)''. ``There are examples of \((Q,\cdot)\) with (LBI) and (\ ) not satisfying \(l_{x,y}=(l_{y,x})^{-1}\) and not \(l_{x,y}=l_{x,y^\cdot(x)}\)''. ``If (R, id) \(\wedge\) (L, id) then: (LLP) \(\Leftrightarrow\) (LBI)''. ``(LLP) \(\wedge\) (R, id) \(\wedge\) \((\forall a\in Q\): \(a^\cdot\circ a^\cdot=(a^\cdot(a))^\cdot\) (left alternative) \(\Rightarrow\) (L, id)''. ``If (LLP) \(\wedge\) (R, id) then: left alternative \(\Leftrightarrow\) (LBI) \(\wedge\) (L, id) (i.e. \((Q,\cdot)\) is a Bol loop)''. ``If \((Q,\cdot)\) is a quasigroup with (LLP) then (R, id)''. Finally the authors study the isotopy of (LLP) quasigroups. The principal isotope of an (LLP) quasigroup is a left Bol loop. If (LLP) then: (R, id) \(\Leftrightarrow (Q,\cdot)\) is a quasigroup.