Wolfram Bentz

Groups synchronizing a transformation of non-uniform kernel

Suppose that a deterministic finite automata A=(Q,@S) is such that all but one letters from the alphabet @S act as permutations of the state set Q and the exceptional letter acts as a transformation with non-uniform kernel. Which properties of the permutation group G generated by the letters acting as permutations ensure that A becomes a synchronizing automaton under every possible choice of the exceptional letter (provided the exceptional letter acts as a transformation of non-uniform kernel)? Such permutation groups are called almost synchronizing. It is easy to see that an almost synchronizing group must be primitive; our conjecture is that every primitive group is almost synchronizing. Clearly every synchronizing group is almost synchronizing. In this paper we provide two different methods to find non-synchronizing, but almost synchronizing groups. The infinite families of examples provided by the two different methods have few overlaps. The paper closes with a number of open problems on group theory and combinatorics.

The rank of the semigroup of transformations stabilising a partition of a finite set

Let

Primitive groups, graph endomorphisms and synchronization

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be a partition of a finite set X . We say that a transformation f : X → X preserves (or stabilises) the partition

Finite affine algebras are fully dualizable

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if for all P ∈

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there exists Q ∈

The Commuting Graph of the Symmetric Inverse Semigroup

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The largest subsemilattices of the endomorphism monoid of an independence algebra

such that Pf ⊆ Q . Let T ( X ,