2. Articole
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Item ON RECURSIVELY DIFFERENTIABLE QUASIGROUPS(2015) Larionova-Cojocaru, IngaIf ( ),Q ⋅ is a binary groupoid then will denote its recursive derivative of order s by „ s ⋅ ”, hence 0 1 2 1 , , , ( ) ( ), s s s x y x y x y y xy x y x y x y − − ⋅ = ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅K for every , .x y Q∈ If the recursive derivatives „ s ⋅ ”, s=1,2,…,k, of a binary quasigroup ( ),Q ⋅ are quasigroup operations, then ( ),Q ⋅ is called recursively k- differentiable. The notions of recursive derivatives and recursively differentiable quasigroups raised in the algebraic coding theory [1]. Recursively differentiable binary quasigroups in particular groups, are studied in the present paper. Proposition 1. If a quasigroup ( ),Q ⋅ , with the left unit, is recursively 1- differentiable then the mapping 2 x x→ is a bijection. Proposition 2. A diassociative loop ( ),Q ⋅ is recursively 1-differentiable if and only if the mapping 2 x x→ is a bijection. Corollary 1. A Moufang loop ( ),Q ⋅ , in particular a group, is recursively 1- differentiable if and only if the mapping 2 x x→ is a bijection on Q . Corollary 2. Finite groups of even order are not recursively 1-differentiable.Item ON RECURSIVE DIFFERENTIABILITY OF BINARY QUASIGROUPS("VALINEX", 2014) Larionova-Cojocaru, Inga; Syrbu, ParascoviaA 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.Item ON INVARIANCE OF RECURSIVE DIFFERENTIABILITY UNDER TH E ISOTOPY OF LEFT BOL LOOPS(CEP USM, 2007) Larionova-Cojocaru, Inga; Syrbu, ParascoviaWe 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.Item ON RECURSIVE DIFFERENTIABILITY OF BINARY QUASIGROUPS(CEP USM, 2015) Larionova-Cojocaru, Inga; Syrbu, ParascoviaRecursively 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.