The Independence Theorem for the congruence lattice and the auto-morphism group of a nite lattice was proved by V. A. Baranski and A. Urquhart. Both proofs utilize the characterization theorem of congruence lattices of nite lattices (as nite distributive lattices) and the characterization theorem of auto-morphism groups of nite lattices (as nite groups). In this paper, we introduce a new, stron...

Among (isotopy classes of) automorphisms of handlebodies those called irreducible (or generic) are the most interesting, analogues of pseudo-Anosov automorphisms of surfaces. We consider the problem of isotoping an irreducible automorphism so that it is most efficient (has minimal growth rate) in its isotopy class. We describe a property, called tightness, of certain invariant laminations, whic...

It is shown that the complete Turing degrees do not form an automorphism base. A class A ⊆ the Turing degrees D is an automorphism base (see Lerman [1983]) if and only if any nontrivial automorphism of D necessarily moves at least one of its elements — or, equivalently, the global action of any such automorphism is completely determined by that on A . Jockusch and Posner [1981] demonstrated the...

We define four families of geometries with as point graph the graph — or its complement — of all elliptic 4 hyperplanes of a given parabolic quadric in any finite 6-dimensional projective space, where adjacency is 5 given by intersecting in a tangent 4-space. One of the classes consists of semi-partial geometries constructed 6 in J.A. Thas [SPG-reguli and semipartial geometries, Adv. Geom. 1 (2...

We associate a 2-complex to the following data: a presentation of a semigroup S and a transitive action of S on a set V by partial transformations. The automorphism group of the action acts properly discontinuously on this 2-complex. A sufficient condition is given for the 2-complex to be simply connected. As a consequence we obtain simple topological proofs of results on presentations of Schüt...

We study the time complexity of McKay’s algorithm to compute canonical forms and automorphism groups of graphs. The algorithm is based on a type of backtrack search, and it performs pruning by discovered automorphisms and by hashing partial information of vertex labelings. In practice, the algorithm is implemented in the nauty package. We obtain colorings of Fürer’s graphs that allow the algori...

We introduce two generalizations of the full inverse symmetric semigroup ISX — the inverse semigroups PIPX and P̃IPX . Both of these semigroups have the same set of elements and contain ISX . The binary operations on PIPX and P̃IPX are two natural generalizations of the multiplication of partial permutations and each of the considered operations coincides with this multiplication when restricted ...

We prove that IHSA, the theory of infinite dimensional Hilbert spaces equipped with a generic automorphism, is א0-stable up to perturbation of the automorphism, and admits prime models up to perturbation over any set. Similarly, APrA, the theory of atomless probability algebras equipped with a generic automorphism is א0-stable up to perturbation. However, not allowing perturbation it is not eve...

Tree-like partial cubes were introduced in [B. Brešar, W. Imrich, S. Klavžar, Tree-like isometric subgraphs of hypercubes, Discuss. Math. Graph Theory, 23 (2003), 227–240] as a generalization of median graphs. We present some incorrectnesses from that article. In particular we point to a gap in the proof of the theorem about the dismantlability of the cube graph of a tree-like partial cube and ...

Baumslag conjectured in the 1970s that the automorphism tower of a finitely generated free group (free nilpotent group) must be very short. Dyer and Formanek [9] justified the conjecture concerning finitely generated free groups in the “sharpest sense” by proving that the automorphism group Aut(Fn) of a non-abelian free group Fn of finite rank n is complete. Recall that a group G is said to be ...