### Abstract:

If ( ),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.

### Description:

LARIONOVA, Inga. On recursively differentiable quasigroups. In: Tendinţe contemporane ale dezvoltării ştiinţei: viziuni ale tinerilor cercetători: conferința științifică internațională a doctoranzilor, 10 martie 2015, Chișinău. Chișinău: Artpoligraf, 2015, p. 21.