Facultatea de Matematică şi Informatică / Faculty of Methematics and Informatics

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Start date: 2014Subject: search.filters.subject.middle Bol loop

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    MIDDLE BRUCK LOOPS AND THE TOTAL MULTIPLICATION GROUP
    (Springer, 2022) Drapal, Ales; Syrbu, Parascovia
    Let 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.
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    LOOPS WITH INVARIANT FLEXIBILITY UNDER THE ISOSTROPHY
    (Institutul de Matematică şi Informatică al AŞM, 2020) Syrbu, Parascovia; Grecu, Ion
    The 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.
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    PSEUDO-AUTOMORPHISMS OF MIDDLE BOL LOOPS
    (CEP USM, 2017) Grecu, Ion
    The set of Moufang elements in a middle Bol loop is considered in the present work. We prove that every inner mapping of the Moufang part (which is a subloop) of a middle Bol loop (Q,·) extends to a right pseudo-automorphism of (Q,·)
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    ON PSEUDOAUTOMORPHISMS OF MIDDLE BOL LOOPS
    (CEP USM, 2015) Grecu, Ion
    A loop (Q1▪) is called a middle Bol loop if every loop isotope of (Q1▪) satisfies the identity (x∙y)-1 = y -1 x -1 (i.e. if the anti-automorphic inverse property is universal in (Q1▪) [1]. Middle Bol loops are isostrophes of left (right) Bol loops [2, 4]. The left (right, middle) pseudoautomorphisms of middle Bol loops are considered in the present article. The general form of middle Bol loop’s autotopisms is given using right pseudoautomorphisms of the corresponding right Bol loops. Necessary and sufficient conditions when a LP-isotope of a middle Bol loop (Q1▪) is isomorphic to (Q1▪) are proved. It is shown that in the left (right) Bol loops every middle seudoautomorphism is a left (right) pseudoautomorphism. Connections between the groups of pseudoautomorphisms (left, right, middle) of a middle Bol loop and of the corresponding left Bol loop are found.
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    ON MULTIPLICATION OF ISOSTROPHIC QUASIGROUPS
    (CEP USM, 2014) Grecu, Ion
    A loop is called a middle Bol loop if every loop isotope of satisfies the identity(i.e. if the anti- autmorphic inverse property is universal in ). Middle Bol loops are isostrophes of left (right) Bol loops. Multiplication groups of a quasigroup and ofloops which areisostrophic to are haracterized. In particular, it is proved that the right multiplication group of a middle Bol loop coincides with the left (right) multiplication group of the corresponding right (left) Bol loop. Some properties of the stabilizer of an element in the right (left) multiplication group of sostrophic quasigroups are established.