Facultatea de Matematică şi Informatică / Faculty of Methematics and Informatics
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Item MIDDLE BRUCK LOOPS AND THE TOTAL MULTIPLICATION GROUP(Springer, 2022) Drapal, Ales; Syrbu, ParascoviaLet Q be a loop. The mappings x↦ ax, x↦ xa and x↦ a/ x are denoted by La, Ra and Da, respectively. The loop is said to be middle Bruck if for all a, b∈ Q there exists c∈ Q such that DaDbDa= Dc. The right inverse of Q is the loop with operation x/ (y\ 1). It is proved that Q is middle Bruck if and only if the right inverse of Q is left Bruck (i.e., a left Bol loop in which (xy) - 1= x- 1y- 1). Middle Bruck loops are characterized in group theoretic language as transversals T to H≤ G such that ⟨ T⟩ = G, TG= T and t2= 1 for each t∈ T. Other results include the fact that if Q is a finite loop, then the total multiplication group⟨ La, Ra, Da; a∈ Q⟩ is nilpotent if and only if Q is a centrally nilpotent 2-loop, and the fact that total multiplication groups of paratopic loops are isomorphic.Item LOOPS WITH INVARIANT FLEXIBILITY UNDER THE ISOSTROPHY(Institutul de Matematică şi Informatică al AŞM, 2020) Syrbu, Parascovia; Grecu, IonThe question ”Are the loops with universal (i.e. invariant under the isotopy of loops) flexibility law xy·x=x·yx , middle Bol loops?” is open in the theory of loops. If this conjecture is true then the loops for which a ll isostrophic loops are flexible are Moufang loops. In the present paper we prove that commutative loops with invariant flexibility under the isostrophy of loops are Mouf ang loops. In particular,we obtain that commutative IP -loops with universal flexibility are Moufang loops.