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This paper proves that the variety generated by a centrally nilpotent Moufang loop (or centrally nilpotent A-loop) is finitely based., Comment: 26 pages

This paper is a natural continuation of paper "On rectifiable spaces and its algebraical equivalents, topological algebraic systems and Mal'cev algebras" published in arxiv:1309.4572. Thus we justify the need to present the entire material in an unified manner. This paper is the continuation of Section 6 from the first paper. It specifies and corrects the roughest mistakes, incorrect statements and nonsense of the introduced concepts, which are available in numerous papers on topological algebraic systems, basically in papers of Academician Choban M. M. and his disciples., Comment: 36 pages

We investigate the rectifiable spaces, the Mal'cev algebras, the almost quasivarieties of topological algebraic systems and their free systems and others. It specifies and corrects the roughest mistakes, incorrect statements and nonsense of the introduced concepts connected with the concepts listed before, which are available in numerous papers on topological algebraic systems, basically in papers of Academician Choban M.M. and his disciples., Comment: 90 pages

The paper defines the notion of alternative loop algebra F[Q] for any nonassociative Moufang loop Q as being any non-zero homomorphic image of the loop algebra FQ of a loop Q over a field F. For the class M of all nonassociative alternative loop algebras F[Q] and for the class L of all nonassociative Moufang loops Q are defined the radicals R and S, respectively. Moreover, for classes M, L is proved an analogue of Wedderburn Theorem for finite dimensional associative algebras. It is also proved that any Moufang loop Q from the radical class R can be embedded into the loop of invertible elements U(F[Q])of alternative loop algebra F[Q]. The remaining loops in the class of all nonassociative Moufang loops L cannot be embedded into loops of invertible elements of any unital alternative algebras., Comment: 47 pages

The structure of the commutative Moufang loops (CML) with minimum condition for subloops is examined. In particular it is proved that such a CML $Q$ is a finite extension of a direct product of a finite number of the quasicyclic groups, lying in the centre of the CML $Q$. It is shown that the minimum conditions for subloops and for normal subloops are equivalent in a CML. Moreover, such CML also characterized by different conditions of finiteness of its multiplication groups., Comment: 18 pages

Let $C(F)$ be a matrix Cayley-Dickson algebra over field $F$. By $M_0(F)$ we denote the loop containing of all elements of algebra $C(F)$ with norm 1. It is shown in this paper that with precision till isomorphism the loops $M_0(F)/<-1>$ they and only they are simple non-associative Moufang loops, where $F$ are subfields of algebraic closed field, Comment: 8 pages

It is known that with precision till isomorphism that only and only loops $M(F) = M_0(F)/<-1>$, where $M_0(F)$ denotes the loop, consisting from elements of all matrix Cayley-Dickson algebra $C(F)$ with norm 1, and $F$ be a subfield of arbitrary fixed algebraically closed field, are simple non-associative Moufang loops. In this paper it is proved that the simple loops $M(F)$ they and only they are not embedded into a loops of invertible elements of any unitaly alternative algebras if $\text{char} F \neq 2$ and $F$ is closed under square root operation. For the remaining Moufang loops such an embedding is possible. Using this embedding it is quite simple to prove the well-known finding: the finite Moufang $p$-loop is centrally nilpotent., Comment: 15 pages