Browsing by Author "Grecu, Ion"

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ASUPRA GRUPULUI MULTIPLICATIV AL BUCLELOR MEDII BOL
(CEP USM, 2014) Grecu, Ion
COMMUTANTS OF MIDDLE BOL LOOPS
(Academy of Sciences of Moldova, 2014) Grecu, Ion; Syrbu, Parascovia
The commutant of a loop is the set of all its elements that commute with each element of the loop. It is known that the commutant of a left or right Bol loop is not a subloop in general. Below we prove that the commutant of a middle Bol loop is an AIP-subloop, i.e., a subloop for which the inversion is an automorphism. A necessary and su cient condition when the commutant is invariant under the existing isostrophy between middle Bol loops and the corresponding right Bol loops is given.
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.
ON MULTIPLICATION GROUPS OF ISOSTROPHIC QUASIGROUPS.
(Valines SRL, 2014) Grecu, Ion
Relations between the multiplication groups of loops which are isostrophes of quasigroups are studied in the present work. We prove that, if (Q; ¢) is a quasigroup and its isostrophe (Q; ±), where x ± y = Ã(y) n '(x), 8x; y 2 Q, is a loop, then the right multiplication group of (Q; ±) is a subgroup of the left multiplica- tion group of (Q; ¢). Moreover, if ' 2 Aut(Q; ±), then RM(Q; ±) is a normal subgroup of LM(Q; ¢). As a corollary from this result we get that the right multiplication group of a middle Bol loop coincides with the left multiplication group of the corresponding right Bol loop.
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.
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.
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,·)