Browsing by Author "Syrbu, Parascovia"

Now showing 1 - 15 of 15

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.
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.
ON A GENERALIZATION OF THE INNER MAPPING GROUP
(CEP USM, 2017) Syrbu, Parascovia
We consider the group GM(Q,·), generated by all left, rightand middle translations of a loop (Q,·). The generalized innermapping groupJ consists of all mappingsα∈GM(Q,·), such that α(e) =e, wheree is the unit of (Q,·). In the present note we give a set of mappings which generates the groupJ.
ON INVARIANCE OF RECURSIVE DIFFERENTIABILITY UNDER TH E ISOTOPY OF LEFT BOL LOOPS
(CEP USM, 2007) Larionova-Cojocaru, Inga; Syrbu, Parascovia
We prove that the recursive derivatives of order 1 of isotopic left Bol loops are isotopic and that every loop, isotopic to a recursively 1-differentiable left Bol loop, is recursively 1-differentiable.The recursive differentiability of di-associative loops is also considered.
ON ORTHOGONAL SYSTEMS OF TERNARY QUASIGROUPS ADMITTING NONTRIVIAL PARATOPIES.
(Institutul de Matematică şi Informatică al AŞM, 2017) Syrbu, Parascovia; Ceban, Dina
In the present workwe describ eallorthogonal systems consisting of three ternary quasigroupop erations and of all (three) ternary selectors,admitting at least one nontrivial paratopy. In[11]we proved that there exist precisely 48 orthogonal systems of the considered form,admitting atleast one paratopy,which components are three quasigroupop erations,or two quasigroupop erations and a selector. Now we show that there exist precisely 105 such systems, admitting at least one nontrivial paratopy which comp onents are two selectors and a quasigroup op eration, or three selectors.
ON QUASIGROUPS WITH SOME MINIMAL IDENTITIES
(CEP USM, 2015) Ceban, Dina; Syrbu, Parascovia
Quasigroups with two identities (of typesT1 and T2) from Belousov-Bennett classification are considered. It is proved that a π-quasigroup of type T2 is also of type only if it satisfies the identity yx∙x=y (the “right keys law”), so π -quasigroups that are of both types T1and T2 are RIP - quasigroups. Also, it is proved that π-quasigroups of type T2 are isotopic to idempotent quasigroups. Necessary and sufficient conditions when a π -quasigroup of typeT2 is isotopic to a group (an abelian group) are found. It is shown that the set of all π-quasigroups of type T2 isotopic to abelian groups is a subvariety in the variety of all π -quasigroups of type T2 and that π- T-quasigroups of type T2 are medial quasigroups. Using the symmetric group on Q∙Q, some considerations for the spectrum of finite π-quasigroups (Q1) of type T1 are discussed.
ON RECURSIVE DIFFERENTIABILITY OF BINARY QUASIGROUPS
("VALINEX", 2014) Larionova-Cojocaru, Inga; Syrbu, Parascovia
A quasigroup is called recursively n-differentiable if its first n recursive derivatives are quasigroups. The class of recursively differential quasigroups is arisen in the theory of MDS codes, in early 2000. Connections between recursive derivatives of different order are found in the present work. It is shown that isomorphic quasigroups have isomorphic recursive derivatives of any order. Also, it is proved that, if the recursive derivative of order one of a finite quasigroup (Q, ·) is commutative, then its group of inner mappings is a subgrup of the group of inner mappings of (Q, ·), of the same index as their corresponding multiplication groups.
ON RECURSIVE DIFFERENTIABILITY OF BINARY QUASIGROUPS
(CEP USM, 2015) Larionova-Cojocaru, Inga; Syrbu, Parascovia
Recursively differentiable binary quasigroups and loops, are considered in the present paper. Invariants under the recursive differentiability of binary quasigroups are found. It is shown that the recursive derivative of order one of an LIP-loop (Q;) (in particular of a left Bol loop) is an isostrophe of its core. The recursively 1-differentiable left Bol loops are studied. Some properties of recursive derivatives of order one of left Bol loops are established.
ON RECURSIVELY DIFFERENTIABLE K-QUASIGROUPS
(2022) Syrbu, Parascovia; Cuznețov, Elena
Recursive differentiability of linear k-quasigroups (k ≥ 2) is studied in the present work. A k-quasigroup is recursively r-differentiable (r is a natu- ral number) if its recursive derivatives of order up to r are quasigroup operations. We give necessary and sufficient conditions of recursive 1-differentiability (respectively, r-differentiability) of the k-group (Q, B), where B(x1, ..., xk) = x1 · x2 · ... · xk, ∀x1, x2, ..., xk ∈ Q, and (Q, ·) is a finite binary group (respectively, a finite abelian binary group). The second result is a generalization of a known criterion of recursive r-differentiability of finite binary abelian groups [4]. Also we consider a method of construction of recursively r-differentiable finite binary quasigroups of high order r. The maximum known values of the parameter r for binary quasigroups of order up to 200 are presented.
On Self-Orthogonal n-ary Quasigroups [Articol]
(2024) Syrbu, Parascovia; Rotari, Tatiana
We consider self-orthogonal finite n-ary quasigroups and give some estimations of their spectrum in the present work. A method of construction of self-orthogonal n-quasigroups, using self-orthogonal quasigroups of lower arity, is given. In particular, it is shown that there exist: 1) self-orthogonal 2k-ary quasigroups of every order q > 3; q ≠ 6, where k ≥ 1; 2) self-orthogonal pk quasigroups of prime order p, for every p ≥ 3 and every k ≥ 1; 3) self-orthogonal 2n-quasigroups of order q, for every q > 3; q ≠ 6 and n + 1 ≢ 0 (mod q).
ON THE HOLOMORPH OF ¼-QUASIGROUPS OF TYPE T1
("VALINEX", 2014) Ceban, Dina; Syrbu, Parascovia
Quasigroups satisfying the identity x¢(x¢(x¢y)) = y are called ¼-quasigroups of type T1. Necessary and su±cient conditions for the holomorph of a ¼-quasigroup of type T1 to be a ¼-quasigroup of type T1 are established. Also, it is proved that the left (right) multiplication group of a ¼-quasigroup of type T1 is isomorphic to some normal subgroup of the left (right) multiplication group of its holomorph, respectively.
ON π-QUASIGROUPS ISOTOPIC TO ABELIAN GROUPS
(Institutul de Matematică şi Informatică al AŞM, 2009) Syrbu, Parascovia
Aπ-quasigroup is a quasigroup satisfying one of the seven minimalidentities from the V.Belousov’s classification given in [1]. Some general results aboutπ-quasigroups isotopic to groups are obtained by V. Belousovand A. Gwaramija in [1]and [2].π-Quasigroups isotopic to abelian groups are investigated in this paper
ON Π-QUASIGROUPS OF TYPET1
(Academy of Sciences of Moldova, 2014) Syrbu, Parascovia; Ceban, Dina
Quasigroups satisfying the identityx(x· xy ) = y are called π -quasigroups of type T 1 . The spectrum of the defining identity is precisely q = 0 or 1(mod 3), except for q = 6. Necessary conditions when a finite π -quasigroup of type T 1 has the order q = 0 (mod 3), are given. In particular, it is proved that a finite π -quasigroup of type T 1 such that the order of its inner mapping group is not divisible by three has a left unit. Necessary and sufficient conditions when the identityx ( x · xy ) = y is invariant under the isotopy of quasigroups (loops) are found
VALENTIN BELOUSOV (20.02.1925 – 23.07.1988)
(Institutul de Matematică şi Informatică al AŞM, 2016) Choban, Mitrofan; Izbash, Vladimir; Şcerbacov, Victor; Syrbu, Parascovia