Related papers: Lattice initial segments of the Turing degrees

Selman's Theorem in classical Computability Theory gives a characterization of the enumeration reducibility for arbitrary sets in terms of the enumeration reducibility on the total sets: $A \le_e B \iff \forall X [X \equiv_{e} X \oplus…

We classify completely the surfaces of general type whose canonical map is 3-to-1 onto a surface of minimal degree in projective space. These surfaces fall into 5 distinct classes and we give explicit examples belonging to each of these…

We introduce the notion of the second lattice width of a lattice polytope and use this to classify lattice triangles by their width and second width. This is equivalent to classifying lattice triangles contained in a given rectangle (and no…

We investigate similarity classes of arithmetic lattices in the plane. We introduce a natural height function on the set of such similarity classes, and give asymptotic estimates on the number of all arithmetic similarity classes,…

Let $G$ be a real centre-free semisimple Lie group without compact factors. I prove that irreducible lattices in $G$ are rigid under two types of sublinear distortions. The first result is that the class of lattices in groups that do not…

In this paper, we introduce and study the Dirichlet series enumerating (proper) equivalence classes of full rank subforms/sublattices of a given quadratic form/lattice, focusing on the positive definite binary case. We obtain formulas…

We prove that the lattice of ideals of an arbitrary $L$-algebra is distributive. As a consequence, a spectral theory applies with no restriction. We also study the spectrum (i.e. the set of prime ideals) of $L$-algebras and characterize…

This paper deals with join-semilattices whose sections, i.e. principal filters, are pseudocomplemented lattices. The pseudocomplement of a\vee b in the section [b,1] is denoted by a\rightarrow b and can be considered as the connective…

We study algebras k[x_1,...,x_n]/I which admit a grading by a subsemigroup of N^d such that every graded component is a one-dimensional k-vector space. V.I.~Arnold and coworkers proved that for d = 1 and n <= 3 there are only finitely many…

The sublattice symmetry on a bipartite lattice is commonly regarded as the chiral symmetry in the AIII class of the tenfold Altland-Zirnbauer classification. Here, we reveal the spatial nature of sublattice symmetry, and show that this…

We investigate issues surrounding an old question of Yates' as to the existence of a minimal Turing degree with no strong minimal cover, specifically with respect to the hyperimmune-free degrees.

In order to study the structure of arbitrary split Leibniz triple systems, we introduce the class of split Leibniz triple systems as the natural extension of the class of split Lie triple systems and split Leibniz algebras. By developing…

The automorphism groups of integral Lorentzian lattices act by isometries on hyperbolic space with finite covolume. In the case of reflective integral lattices, the automorphism groups are commensurable to arithmetic hyperbolic reflection…

We show that many important varieties and sets of varieties of semigroups may be defined by relatively simple and transparent first-order formulas in the lattice of all semigroup varieties.

Let $A$ be a semigroup whose only invertible element is 0. For an $A$-homogeneous ideal we discuss the notions of simple $i$-syzygies and simple minimal free resolutions of $R/I$. When $I$ is a lattice ideal, the simple 0-syzygies of $R/I$…

In the current paper we study extremal semilattices with respect to their equational properties. In the class $\mathbf{S}_n$ of all semilattices of order $n$ we find semilattices which have maximal (minimal) number of consistent equations.…

In this paper we study prime, maximal and two--class congruences from the point of view of the relationships between them in various kinds of universal algebras, as well as their direct and inverse images through morphisms. This research…

Let A be an associative algebra with identity over a field k. An atomistic subsemiring R of the lattice of subspaces of A, endowed with the natural product, is a subsemiring which is a closed atomistic sublattice. When R has no zero…

We establish several independent results concerning extremal, left modular, congruence uniform, and semidistributive lattices. An equivalent characterization of left modular lattices is obtained in terms of edge-labellings, together with…

In this expository article, we give a self-contained introduction to the wonderfully well-behaved class of pseudocompact algebras, focusing on the foundational classes of semisimple and separable algebras. We give characterizations of such…