Parascovia Sîrbu
Moldova State University, Chişinău, Moldova

The multiplication group (total multiplication group) of a loop $(Q,\cdot )$ is the group generated by all its right and left translations (respectively, by all its right, left and middle translations). We investigate the total multiplication groups of middle Bol loops, i.e. of loops satisfying the middle Bol identity $x(yz\setminus x)=(x/z)(y\setminus x)$. The last identity is the necessary and sufficient condition when anti-automorphic inverse property is invariant under the isotopy of loops [1]. It is proved that total multiplication groups of isostrophic loops are isomorphic. Commutative middle Bol loops are characterized [2]. An open problem in the theory of loops is if the loops with invariant flexibility ($x\cdot yx = xy\cdot x$), under the isotopy of loops, are middle Bol loops [3]. If this conjecture is true than the loops with invariant middle Bol identity under the isostrophy of loops are Moufang loops. A necessary and sufficient condition when the middle Bol identity is invariant under the isostrophy of loops is given in [4], and it is proved that commutative loops with invariant flexibility under the isostrophy of loops are Moufang loops.

[1] Belousov V., Foundations of the theory of quasigroups and loops (Russian). Nauka, Moscow, 1967
[2] Drapal, A., Syrbu, P., Middle Bruck Loops and the Total Multiplication Group, Results Math 77, 174 (2022)
[3] Syrbu P., Loops with universal elasticity, Quasigroups and Related Systems 1 (1), (1994), 57-65
[4] Syrbu P., Grecu I., On loops with invariant flexibility under isostrophy, Bul. Acad. Stiinte Repub. Moldova, Mat., 2020, No. 1(92), 122-128