A loop $Q$ is said to be left conjugacy closed if the set $A=\{L_x/x\in Q\}$ is closed under conjugation. Let $Q$ be an LCC loop, let $ℒ$ and $ℛ$ be the left and right multiplication groups of $Q$ respectively, and let $I\left(Q\right)$ be its inner mapping group, $M\left(Q\right)$ its multiplication group. By Drápal’s theorem [3, Theorem 2.8] there exists a homomorphism $\Lambda :ℒ\to I\left(Q\right)$ determined by $L_x\to R_x^{-1}{L}_x$. In this short note we examine different possible extensions of this $\Lambda$ and the uniqueness of these extensions.

Let $Q$ be a loop. If $S\le Q$ is such that $\varphi \left(S\right)\subseteq S$ for each standard generator of  Inn$Q$, then $S$ does not have to be a normal subloop. In an LC loop the left and middle nucleus coincide and form a normal subloop. The identities of Osborn loops are obtained by applying the idea of nuclear identification, and various connections of Osborn loops to Moufang and CC loops are discussed. Every Osborn loop possesses a normal nucleus, and this nucleus coincides with the left, the right and the middle nucleus....

It is well-known that a based space is of the weak homotopy type of a loop space iff it is a grouplike algebra over an $A_{\infty }$-operad. The classical model for such an operad consists of Stasheff’s associahedra. The present paper describes a similar recognition principle for free loop spaces. Let $𝒫$ be an operad, $M$ a $𝒫$-module and $U$ a $𝒫$-algebra. An $M$-trace over $U$ consists of a space $V$ and a module homomorphism $T:M\to {\text{End}}_{U,V}$ over the operad homomorphism $𝒫\to {\text{End}}_U$ given by the algebra structure on $U$. Let ${𝒞}_1$ be the...

Let $G$ be a finite group and $C_2$ the cyclic group of order $2$. Consider the $8$ multiplicative operations $(x,y)\mapsto {\left(x^i{y}^j\right)}^k$, where $i$, $j$, $k\in \{-1,1\}$. Define a new multiplication on $G\times C_2$ by assigning one of the above $8$ multiplications to each quarter $(G\times \{i\left\}\right)\times (G\times \{j\left\}\right)$, for $i,j\in C_2$. We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovsk’y: On the uniqueness of loops $M(G,2)$.