Narrow Results

Strongly nil-cleanness and uniquely nil-cleanness are two important clean properties of rings. The aim of this paper is to answer when a semigroup ring is strongly or uniquely nil-clean. We obtain conditions on a coefficient ring together with conditions on a semigroup that completely characterize when the corresponding semigroup ring is strongly nil-clean. Moreover, when the coefficient ring is an integral domain, we characterize when the corresponding semigroup ring is uniquely nil-clean.

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Recently, several infinite families of congruences modulo 12, 27 and 40 for $\overline{p}(n)$ have been established. In this paper, we investigate systematically some other infinite families of congruences modulo 12, 24 and 40 for $\overline{p}(n)$.

In this paper, we describe the set of prime numbers splitting completely in the non-abelian splitting field of certain monic irreducible polynomials of degree $3.$ As an application, we establish some divisibility properties of the associated ternary recurrence sequence by primes $p$, thus greatly extending recent work of Evink and Helminck [Tribonacci numbers and primes of the form $p=x^2+11y^2$, Math. Slovaca69(3) (2019) 521–532] and of Faisant [On the Padovan sequence (2019), https://arxiv.org/abs/1905.07702]. We also prove some new results on the number of solutions of the characteristic equation of the recurrence sequence modulo $p$.

The largest common point set problem (LCP) is, given two point set P and Q in d-dimensional Euclidean space, to find a subset of P with the maximum cardinality that is congruent to some subset of Q. We consider a special case of LCP in which the size of the largest common point set is at least (|P| + |Q| - k)/2. We develop efficient algorithms for this special case of LCP and a related problem. In particular, we present an O(k³n^1.34 + kn²log n) time algorithm for LCP in two-dimensions, which is much better for small k than an existing O(n^3.2log n) time algorithm, where n = max{|P|,|Q|}.

This paper presents an algorithm that tests the congruence of two sets of n points in d-dimensional space in time. This improves the previous best algorithm for dimensions d≥6.

Let $P$ and $Q$ be two discrete point sets in ${ℝ}^d$ of sizes $m$ and $n$, respectively, and let $𝜖>0$ be a given input threshold. The largest common point set problem (LCP) seeks the largest subsets $A\subseteq P$ and $B\subseteq Q$ such that $\left|A\right|=\left|B\right|$ and there exists a transformation $\varphi$ that makes the bottleneck distance between $\varphi (A)$ and $B$ at most $𝜖$. We present two algorithms that solve a relaxed version of this problem under translations in ${ℝ}^d$ and under rigid motions in the plane, and that takes an additional input parameter $\eta >0$. Let $\ell$ be the largest subset size achievable for the given $𝜖$. Our algorithm finds subsets $A\subseteq P$ and $B\subseteq Q$ of size $\left|A\right|=\left|B\right|\ge \ell$ and a transformation $\varphi$ such that the bottleneck distance between $\varphi (A)$ and $B$ is at most $(1+\eta )𝜖$. For rigid motions in the plane, the running time is $O(\frac{n^2{m}^2}{\eta }(n+m)logn)$. For translations in ${ℝ}^d$, the running time is $O(\frac{nm}{{\eta }^{\kappa }}{(n+m)}^{1.5}logn)$, where $\kappa =1$ for $d=2$ and $\kappa =2d-1$ for $d\ge 3$.

The Congruence Lattice Problem (CLP), stated by R. P. Dilworth in the forties, asks whether every distributive {∨, 0}-semilattice S is isomorphic to the semilattice Con_c L of compact congruences of a lattice L.

While this problem is still open, many partial solutions have been obtained, positive and negative as well. The solution to CLP is known to be positive for all S such that |S|≤ℵ₁. Furthermore, one can then take Lwith permutable congruences. This contrasts with the case where |S|≥ℵ₂, where there are counterexamples S for which Lcannot be, for example, sectionally complemented. We prove in this paper that the lattices of these counterexamples cannot have permutable congruences as well.

We also isolate finite, combinatorial analogues of these results. All the "finite" statements that we obtain are amalgamation properties of the Con_c functor. The strongest known positive results, which originate in earlier work by the first author, imply that many diagrams of semilattices indexed by the square2² can be lifted with respect to the Con_c functor.

We prove that the latter results cannot be extended to the cube, 2³. In particular, we give an example of a cube diagram of finite Boolean semilattices and semilattice embeddings that cannot be lifted, with respect to the Con_c functor, by lattices with permutable congruences.

We also extend many of our results to lattices with almost permutable congruences, that is, α∨β=αβ ∪βα, for all congruences α and β.

We conclude the paper with a very short proof that no functor from finite Boolean semilattices to lattices can lift the Con_c functor on finite Boolean semilattices.

We construct a diagram , indexed by a finite partially ordered set, of finite Boolean 〈∨, 0, 1〉-semilattices and 〈∨, 0, 1〉-embeddings, with top semilattice 2⁴, such that for any variety V of algebras, if has a lifting, with respect to the congruence lattice functor, by algebras and homomorphisms in V, then there exists an algebra U in V such that the congruence lattice of U contains, as a 0,1-sublattice, the five-element modular nondistributive lattice M₃. In particular, V has an algebra whose congruence lattice is neither join- nor meet-semidistributive Using earlier work of K. A. Kearnes and Á. Szendrei, we also deduce that V has no nontrivial congruence lattice identity.

In particular, there is no functor Φ from finite Boolean semilattices and 〈∨, 0, 1〉-embeddings to lattices and lattice embeddings such that the composition Con Φ is equivalent to the identity (where Con denotes the congruence lattice functor), thus solving negatively a problem raised by P. Pudlák in 1985 about the existence of a functorial solution of the Congruence Lattice Problem.

For a class of algebras, denote by Con_c the class of all (∨, 0)-semilattices isomorphic to the semilattice Con_cA of all compact congruences of A, for some A in . For classes and of algebras, we denote by the smallest cardinality of a (∨, 0)-semilattices in Con_c which is not in Con_c if it exists, ∞ otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruence-distributive varieties and , is either finite, or ℵ_n for some natural number n, or ∞. We also find two finitely generated modular lattice varieties and such that , thus answering a question by J. Tůma and F. Wehrung.

We consider the problem, whether the algebras in two finitely generated congruence-distributive varieties have isomorphic congruence lattices. According to the results of P. Gillibert, this problem is closely connected with the question, which diagrams of finite distributive semilattices can be represented by the congruence lattices of algebras in a given variety. We study this question for varieties of bounded lattices, generated by different nondistributive lattices of length 2 (denoted M_n). For each pair from this family of varieties we construct a diagram indexed by the product of three finite chains, which is liftable in one variety and nonliftable in the other one. We also discover an interesting link to the four-color theorem of graph theory.

A quasivariety is a universal Horn class of algebraic structures containing the trivial structure. The set of all subquasivarieties of a quasivariety forms a complete lattice under inclusion. A lattice isomorphic to for some quasivariety is called a lattice of quasivarieties or a quasivariety lattice. The Birkhoff–Maltsev Problem asks which lattices are isomorphic to lattices of quasivarieties. A lattice L is called unreasonable if the set of all finite sublattices of L is not computable, that is, there is no algorithm for deciding whether a finite lattice is a sublattice of L. The main result of this paper states that for any signature σ containing at least one non-constant operation, there is a quasivariety of signature σ such that the quasivariety lattice is unreasonable. Moreover, there are uncountable unreasonable lattices of quasivarieties. We also present some corollaries of the main result.

We prove that the class K(σ) of all algebraic structures of signature σ is Q-universal if and only if there is a class K ⊆ K(σ) such that the problem whether a finite lattice embeds into the lattice of K-quasivarieties is undecidable.

We find sufficient conditions guaranteeing that for a quasivariety $M$ of structures of finite type containing a $\mathrm{\text{B}}$-class with respect to $M$, there exists a subquasivariety $K\subseteq M$ and a structure ${𝒜}\in K$ such that the problems whether a finite lattice embeds into the lattice $\mathrm{\text{Lv}}(K)$ of $K$-varieties and into the lattice ${Con}_K{𝒜}$ are undecidable.

It is well known that an equivalence relation is invariant under the basic operations of an algebra if and only if it is invariant under the unary polynomials of the algebra. We show that a higher arity version of this property holds for a higher dimensional analogue of an equivalence relation. It follows that the $(n)$-ary hypercommutator for an algebra is determined by its $(n)$-ary polynomials. We construct examples to show that this fails for every arity of the term condition higher commutator.

In the previous paper [K. S. Lau, J. J. Luo and H. Rao, Topological structure of fractal squares, Math. Proc. Camb. Phil. Soc.155 (2013) 73–86], Lau, Luo and Rao completely classified the topological structure of so called fractal square$F$ defined by $F=(F+{𝒟})/n$, where ${𝒟}⊊{\{0,1,\dots ,n-1\}}^2,n\ge 2$. In this paper, we further provide simple criteria for the $F$ to be totally disconnected, then we discuss the Lipschitz classification of $F$ in the case $n=3$, which is an attempt to consider non-totally disconnected sets.

It was long assumed that the pseudorandom distribution of prime numbers was free of biases. Specifically, while the prime number theorem gives an asymptotic measure of the probability of finding a prime number and Dirichlet’s theorem on arithmetic progressions tells us about the distribution of primes across residue classes, there was no reason to believe that consecutive primes might “know” anything about each other — that they might, for example, tend to avoid ending in the same digit. Here, we show that the Iterated Function System method (IFS) can be a surprisingly useful tool for revealing such unintuitive results and for more generally studying structure in number theory. Our experimental findings from a study in 2013 include fractal patterns that reveal “repulsive” phenomena among primes in a wide range of classes having specific congruence properties. Some of the phenomena shown in our computations and interpretation relate to more recent work by Lemke Oliver and Soundararajan on biases between consecutive primes. Here, we explore and extend those results by demonstrating how IFS points to the precise manner in which such biases behave from a dynamic standpoint. We also show that, surprisingly, composite numbers can exhibit a notably similar bias.

In this paper, the concept of $U$-compatible congruence and its related properties are studied. We also focus on homomorphism theorems for uninorms. Decomposition of uninorms is investigated via factor congruences, at first. Subsequently, we investigate decomposition of conjunctive and disjunctive uninorms, respectively. After all, we determine necessary and sufficient conditions for decomposition of some uninorms that are neither conjunctive nor disjunctive. In addition to all these, we shed light on the application of our results with the examples.

This paper considers the problem of constructing shape measures; we start by giving a short overview of areas of practical application of such measures.

Shapes can be characterised in terms of a set of geometric properties, some of which are Boolean in nature. We give two general principles for turning Boolean properties into continuous measures of that property. (For example, we may turn the Boolean property conves into a convexity value in the range [0–1].) We demonstrate how these principles can be applied to construct various shape measures, including ones for convexity, circularity, ellipticity, triangularity, rectilinearity, rectangularity and symmetry in two dimensions, and 2.5D-ness, stability, and imperforateness in three dimensions. Some of these measures are new; others are well known and we show how they fit into this general framework.

We also show how such measures for a single shape can be generalised to multiple shapes, and consider as examples measures for containment, resemblance, congruence, and similarity.

For a semimodule over any semiring, the maximal and prime $k$-subsemimodules are characterized with the help of the quotient structure. The main results are as follows. A semimodule is $k$-congruence-simple iff it is $k$-subsemimodule-simple. A nonzero semimodule is $k$-simple iff it satisfies condition $(C)$. A proper $k$-subsemimodule of a semimodule is a maximal $k$-subsemimodule iff the quotient semimodule is $k$-simple. A proper $k$-subsemimodule is prime in a semimodule iff the quotient semimodule is prime. A nonzero $k$-simple semimodule over a commutative semiring is prime. Lemma 6.2 in this paper corrects the statement and proof of Lemma 3.16 in [Yeşilot Hacet. J. Math. Stat.39 (2010) 305–312].

When considering affine tropical geometry, one often works over the max-plus algebra (or its supertropical analog), which, lacking negation, is a semifield (respectively, $\nu$-semifield) rather than a field. One needs to utilize congruences rather than ideals, leading to a considerably more complicated theory. In his dissertation, the first author exploited the multiplicative structure of an idempotent semifield, which is a lattice ordered group, in place of the additive structure, in order to apply the extensive theory of chains of homomorphisms of groups. Reworking his dissertation, starting with a semifield${}^{†}$$F$, we pass to the semifield${}^{†}$$F({\lambda }_1,\dots ,{\lambda }_n)$ of fractions of the polynomial semiring${}^{†}$, for which there already exists a well developed theory of kernels, which are normal convex subgroups of $F({\lambda }_1,\dots ,{\lambda }_n)$; the parallel of the zero set now is the $1$-set, the set of vectors on which a given rational function takes the value 1. These notions are refined in supertropical algebra to $\nu$-kernels (Definition 4.1.4) and $1^{\nu }$-sets, which take the place of tropical varieties viewed as sets of common ghost roots of polynomials. The $\nu$-kernels corresponding to tropical hypersurfaces are the $1^{\nu }$-sets of what we call “corner internal rational functions,” and we describe $\nu$-kernels corresponding to “usual” tropical geometry as $\nu$-kernels which are “corner-internal” and “regular.” This yields an explicit description of tropical affine varieties in terms of various classes of $\nu$-kernels. The literature contains many tropical versions of Hilbert’s celebrated Nullstellensatz, which lies at the foundation of algebraic geometry. The approach in this paper is via a correspondence between $1^{\nu }$-sets and a class of $\nu$-kernels of the rational $\nu$-semifield${}^{†}$ called polars, originating from the theory of lattice-ordered groups. When $F$ is the supertropical max-plus algebra of the reals, this correspondence becomes simpler and more applicable when restricted to principal$\nu$-kernels, intersected with the $\nu$-kernel generated by $F$. For our main application, we develop algebraic notions such as composition series and convexity degree, leading to a dimension theory which is catenary, and a tropical version of the Jordan–Hölder theorem for the relevant class of $\nu$-kernels.