On the structure of left and right F-, SM-, and E-quasigroups

Abstract

Abstract. It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM-and E-quasigroups.Information on simple quasigroups from these quasigroup classes is given, for example, finite simple Fquasigroup is a simple group or a simple medial quasigroup.It is proved that any left F-quasigroup is isotopic to the direct product of a group and a left S-loop. Some properties of loop isotopes of F-quasigroups (including M-loops) are pointed out. A left special loop is an isotope of a left F-quasigroup if and only if this loop is isomorphic the direct product of a group and a left S-loop (this is an answer to Belousov "1a" problem).Any left FESM-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(2) problem). New proofs of some known results on the structure of commutative Moufang loops are presented.2000 Mathematics Subject Classification: 20N05

An analogue of the ElGamal scheme based on the Markovski algorithm

“…1 2 3 0 4 6 5 γ 6 = 0 1 2 3 4 5 6 6 4 1 2 5 0 3 In our example T (5,3,6) we get: α 5 = (0516)(243); β 3 = (0123)(56); γ 6 = (0632145).…”

“…Then Alice's keys are as follows: The private key: m = 3, n = 6, k = 5. The public key is (Q, f ), T, T (3,6,5) = (α 3 , β 6 , γ 5 ) and the Markovski algorithm, where: α 3 = (0615); β 6 = (02)(13); γ 5 = (0315624), γ −5 = (0426513).…”

“…To send a message b = 630512403, Bob computes from the known T = (α, β, γ): α = (234)(0516); β = (0321)(56); γ = (1236054), calculates isotopy T (r,s,t) for random numbers r = 5, s = 3, t = 6, i.e. T (5,3,6) : α 5 = 0 1 2 3 4 5 6 5 6 4 2 3 1 0…”

“…In this algorithm, isostrophy [6] can also be used instead of isotopy, the modified algorithm instead of the Markovski algorithm and n-ary (n > 2) quasigroups [7; 8] instead of binary quasigroups.…”

Bruck decomposition for endomorphisms of quasigroups

“…Recently progress has been made in this topic (cf. [7,8,9]). It is known that in an LF-quasigroup the map e(x) = x\x is an endomorphism, which we call the left deviation.…”