Automorphisms of Jacobian Kummer surfaces

We study automorphisms of a generic Jacobian Kummer surface. First weanalyse the action of classically known automorphisms on the Picard lattice of the surface, then proceed to construct new automorphisms not generated by classical ones. We find 192 such automorphisms, all conjugateby the symmetry group of the (16,6)-configuration.     

Groups of Automorphisms of Tournaments

By the Holland Representation Theorem, every lattice ordered group (l-group) is isomorphic to a subalgebra of the l-group of automorphisms of a chain. Since weakly associative lattice groups (wal-groups) and tournaments are non-transitive generalizations of l-groups and chains, respectively, the problem concerning the possibility of representation of wal-groups by automorphisms of tournaments arises. In the paper we describe the class of wal-groups isomorphic to wal-groups of automorphisms of tournament and show some of its properties.     

Definability for equational theories of commutative groupoids

We find several large classes of equations with the property that every automorphism of the lattice of equational theories of commutative groupoids fixes any equational theory generated by such equations, and every equational theory generated by finitely many such equations is a definable element of the lattice. We conjecture that the lattice has no non-identical automorphisms.     

Dedicated to H. Wielandt on the occasion of his 90th birthday

We determine some relevant structural properties of the group of lattice automorphisms of a nonperiodic modular group.     

Invariant subspaces for translation,dilation and multiplication semigroups

We study invariant subspaces in the context of the work of Katavolos and Power [9] and [10] when one of the semigroups considered is replaced by a discrete one. As a consequence, a rather striking connection is given with the study of the lattice of invariant subspaces of composition operators induced by automorphisms of the unit disc acting on the classical Hardy space. As a particular instance, our study concerns the lattice of invariant subspaces of those composition operators induced by hyperbolic automorphisms, and therefore with the Invariant Subspace Problem. Partially supported by Plan Nacional I+D grant no. MTM2006-06431 and Gobierno de Aragón research group Análisis Matemático y Aplicaciones, ref. DGA E-64.     

Graph Automorphisms and Cells of Lattices

In this paper we apply the notion of cell of a lattice for dealing with graph automorphisms of lattices (in connection with a problem proposed by G. Birkhoff).     

Definability in substructure orderings, I: finite semilattices

We investigate definability in the set of isomorphism types of finite semilattices ordered by embeddability; we prove, among other things, that every finite semilattice is a definable element in this ordered set. Then we apply these results to investigate definability in the closely related lattice of universal classes of semilattices; we prove that the lattice has no non-identical automorphisms, the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets and each element of the two subsets is a definable element in the lattice.     

Relating De Morgan triples with Atanassov’s intuitionistic De Morgan triples via automorphisms

In this paper the relation between De Morgan triples on the unit interval and Atanassov’s intuitionistic De Morgan triples is presented, showing how to obtain, in a canonical way, Atanassov’s intuitionistic De Morgan triples from De Morgan triples. Moreover, we also show that the automorphisms on the unit interval and on L∗ (the intuitionistic value lattice) are in one-to-one correspondence and how automorphisms on L∗ act on Atanassov’s intuitionistic De Morgan triples. It is also proved that the action of automorphisms and the canonical construction of De Morgan triples on L∗ commutes.     

On symmetric algebras of fuzzy sets

In this paper all symmetric algebras of fuzzy sets taking values on a distributive directly nondecomposable lattice with universal bounds 1 and 0, and their classes with respect to automorphisms, are studied. Special attention is devoted to the case where fuzzy sets take values on [0, 1].     

Automorphism groups of algebraic lattices

We classify those algebraic lattices whose group of automorphisms is transitive on the set of elements of the lattice except the smallest and the greatest. We describe their automorphism groups in terms of generalized wreath powers.Presented by Bjarni Jonsson.     

Elliptic K3 Surfaces with Abelian and Dihedral Groups of Symplectic Automorphisms

We analyze K3 surfaces admitting an elliptic fibration ? and a finite group G of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration ?/G comparing its properties to the ones of ?.

We show that if ? admits an n-torsion section, its quotient by the group of automorphisms induced by this section admits again an n-torsion section, and we describe the coarse moduli space of K3 surfaces with a given finite group contained in the Mordell–Weil group.

Considering automorphisms coming from the base of the fibration, we find the Mordell–Weil lattice of a fibration described by Kloosterman, and we find K3 surfaces with dihedral groups as group of symplectic automorphisms. We prove the isometries between lattices described by the author and Sarti and lattices described by Shioda and by Greiss and Lam.     

The Schützenberger group of anH-class in the semigroup of binary relations

K. A. Zaretskii has associated a lattice V(α) with each binary relation α, and he has shown that H_α is isomorphic with the group of all automorphisms of V(α) if H_α is a group. This result is extended in this paper by showing that for any binary relation α, the Schützenberger group Γ (H_α) is isomorphic with the group of all automorphisms of V(α).     

On Deza graphs with triangular and lattice graph complements as parameters

Under consideration are the strictly Deza graphs that are obtained from the complements to triangular and lattice graphs, the Chang graphs, and the Shrikhande graph, by means of their order two automorphisms. It is shown that these graphs are characterized in the class of strictly Deza graphs by the parameters and the structure of neighborhoods.     

Induced residuated maps

The purpose of this paper is to introduce a class of mappings from a lattice L, whose elements are residuated maps, into itself. The main results of this paper identify certain injective residuated mappings of L and order automorphisms of a sublattice of L with mappings from this class.     

Lattice theory and metric geometry

K. Menger and G. Birkhoff recognized 70 years ago that lattice theory provides a framework for the development of incidence geometry (affine and projective geometry). We show in this article that lattice theory also provides a framework for the development of metric geometry (including the euclidean and classical non-euclidean geometries which were first discovered by A. Cayley and F. Klein). To this end we introduce and study the concept of a Cayley–Klein lattice. A detailed investigation of the groups of automorphisms and an algebraic characterization of Cayley–Klein lattices are included. The authors would like to thank an unknown referee for his helpful suggestions.     

Theorie des ecoulements dans les graphes orientes sans circuit

The paper deals with tournaments (i.e., with trichotomic relations) and their homomorphisms. The study of tournaments by means of their homomorphisms is natural as tournaments are algebras of a special kind. We prove (1) theorems which relate combinatorial and algebraic notions (e.g., the score of a tournament and the monoid of its endomorphisms); (2) theorems concerned with strictly algebraic aspects of tournaments (e.g., characterizing the lattice of congruences of a tournament). Our main result is that the group of automorphisms and the lattice of congruences of a tournament are in general independent. In the last part of the paper we give some examples and applications to other fields.     

Definability in the lattice of equational theories of commutative semigroups

In this paper we study first-order definability in the lattice of equational theories of commutative semigroups. In a series of papers, J. Jezek, solving problems posed by A. Tarski and R. McKenzie, has proved, in particular, that each equational theory is first-order definable in the lattice of equational theories of a given type, up to automorphism, and that such lattices have no automorphisms besides the obvious syntactically defined ones (with exceptions for special unary types). He has proved also that the most important classes of theories of a given type are so definable. In a later paper, Jezek and McKenzie have ``almost proved" the same facts for the lattice of equational theories of semigroups. There were good reasons to believe that the same can be proved for the lattice of equational theories of commutative semigroups. In this paper, however, we show that the case of commutative semigroups is different.

     

On Anosov automorphisms of nilmanifolds

The only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds, and the existence of such automorphisms is a really strong condition on the rational nilpotent Lie algebra determined by the lattice, so called an Anosov Lie algebra. We prove that n⊕?⊕n (s times, s≥2) has an Anosov rational form for any graded real nilpotent Lie algebra n having a rational form. We also obtain some obstructions for the types of nilpotent Lie algebras allowed, and use the fact that the eigenvalues of the automorphism are algebraic integers (even units) to show that the types (5,3) and (3,3,2) are not possible for Anosov Lie algebras.     

On fixed points of holomorphic automorphisms

Summary Banach M-lattices are studied from the view point whether all the biholomorphic automorphisms of their unit balls admit fixed points when continuously extended to the closure of the unit ball. A characterization of compact topologieal F-spaces is found in terms of the fixed points of the elements of AutB(C()) which enables to establish some particular properties also of the topological automorphisms of compact F-spaces. Finally it is shown that if the M- lattice E admits a predual then each member of Aut B(E) has fixed point if and only if E is isometrically isomorphic with some l-space.