# A problem in the spectral theory of an ordinary differential by V. A. Tkachenko By V. A. Tkachenko

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Let D be a digraph. We say that V is isomorphically n-complete if the complete transformation semigroup on n letters embeds in the transformation semigroup of D. D is homomorphically n-complete if the full transformation semigroup on n letters divides transformation semigroup of D. D is n-complete (with respect to its semigroup) if the symmetric semigroup on n letters divides the semigroup of D. Now we prove the following statement. 15. Let D be a digraph containing all loop edges. Suppose that V has a strongly connected subdigraph with at least n + 1 vertices which contains a branch.

Products of automata over interconnection digraphs). , minimal generating system. By these negative results we know it is hopeless to seek such bases. In the last part of the chapter we show some simple but important properties of automata products which are also considered automata networks. These include presentations of the well-known classical decomposition theorems ofGluskov and Letichevsky that characterize minimal computational elements that are nevertheless powerful enough for different kinds of computational completeness.

Then there exists in S a subgroup G such that the permutation group G generated by these permutations ofZ is a homomorphic image ofG. It is not difficult to verify the following useful fact. 15. For all finite or infinite transformation semigroups (X, S), ( X ' , S'), (Y, T), and (Y', T'), we have the following: (1) (Y, T) < (Y , T') and (X, S) < (X', S'), then (Y, T) (X, S) < ( Y ' , T') (2) If ( X ' , Sf) is a permutation group and T' contains an idempotent, then (Y', T') and (X, S) ( X ' , S') implies (Y, T) (X, S) (Y', T') (X', (3) For permutation groups, it always holds that if (Y, T) (Y', T') and ( X ' , S'), then (Y, T) (X, S) (Y', T') (X', S').