The concept of a Goldie extending module is generalized to a Goldie extending element in a lattice. An element $a$ of a lattice $L$ with $0$ is said to be a Goldie extending element if and only if for every $b\le a$ there exists a direct summand $c$ of $a$ such that $b\wedge c$ is essential in both $b$ and $c$. Some properties of such elements are obtained in the context of modular lattices. We give a necessary condition for the direct sum of Goldie extending elements to be Goldie extending. Some characterizations...

We investigate the lattice $𝐋\left(𝐕\right)$ of subspaces of an $m$-dimensional vector space $𝐕$ over a finite field $\mathrm{GF}\left(q\right)$ with a prime power $q=p^n$ together with the unary operation of orthogonality. It is well-known that this lattice is modular and that the orthogonality is an antitone involution. The lattice $𝐋\left(𝐕\right)$ satisfies the chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when $𝐋\left(𝐕\right)$ is orthomodular....

Using the geometry of the projective plane over the finite field ${𝔽}_q$, we construct a Hermitian Lorentzian lattice $L_q$ of dimension $(q^2+q+2)$ defined over a certain number ring $𝒪$ that depends on $q$. We show that infinitely many of these lattices are $p$-modular, that is, $p{L}_q^{\text{'}}=L_q$, where $p$ is some prime in $𝒪$ such that ${\left|p\right|}^2=q$. The Lorentzian lattices $L_q$ sometimes lead to construction of interesting positive definite lattices. In particular, if $q\equiv 3mod4$ is a rational prime such that $(q^2+q+1)$ is norm of some element in...

In this paper, we propose the general methods, yielding uninorms on the bounded lattice $(L,\le ,0,1)$, with some additional constraints on $e\in L\setminus \{0,1\}$ for a fixed neutral element $e\in L\setminus \{0,1\}$ based on underlying an arbitrary triangular norm $T_e$ on $[0,e]$ and an arbitrary triangular conorm $S_e$ on $[e,1]$. And, some illustrative examples are added for clarity.

Suppose that $f$ is a primitive Hecke eigenform or a Mass cusp form for ${\Gamma }_0\left(N\right)$ with normalized eigenvalues ${\lambda }_f\left(n\right)$ and let $X>1$ be a real number. We consider the sum $${𝒮}_k\left(X\right):=\sum _{n<X}\sum _{n=n_1,n_2,...,n_k}{\lambda }_f\left(n_1\right){\lambda }_f\left(n_2\right)...{\lambda }_f\left(n_k\right)$$ and show that ${𝒮}_k\left(X\right){\ll }_{f,ϵ}{X}^{1-3/\left(2\right(k+3\left)\right)+ϵ}$ for every $k\ge 1$ and $ϵ>0$. The same problem was considered for the case $N=1$, that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for $k\ge 5$. Since the result is valid...

Let $H$ denote a finite index subgroup of the modular group $\Gamma$ and let $\rho$ denote a finite-dimensional complex representation of $H.$ Let $M\left(\rho \right)$ denote the collection of holomorphic vector-valued modular forms for $\rho$ and let $M\left(H\right)$ denote the collection of modular forms on $H$. Then $M\left(\rho \right)$ is a $\mathbb{Z}$-graded $M\left(H\right)$-module. It has been proven that $M\left(\rho \right)$ may not be projective as a $M\left(H\right)$-module. We prove that $M\left(\rho \right)$ is Cohen-Macaulay as a $M\left(H\right)$-module. We also explain how to apply this result to prove that if $M\left(H\right)$ is a polynomial ring, then...

We give a full description of locally finite $2$-groups $G$ such that the normalized group of units of the group algebra $FG$ over a field $F$ of characteristic $2$ has exponent $4$.

For a t-norm T on a bounded lattice $(L,\le )$, a partial order ${\le }_T$ was recently defined and studied. In [11], it was pointed out that the binary relation ${\le }_T$ is a partial order on $L$, but $(L,{\le }_T)$ may not be a lattice in general. In this paper, several sufficient conditions under which $(L,{\le }_T)$ is a lattice are given, as an answer to an open problem posed by the authors of [11]. Furthermore, some examples of t-norms on $L$ such that $(L,{\le }_T)$ is a lattice are presented.

Let $N$ be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than $4$. Our main theorem is an asymptotic formula solely in terms of $N$ for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves $X_1\left(N\right)/\mathbb{Q}$. From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian $J_1\left(N\right)/\mathbb{Q}$ of $X_1\left(N\right)/\mathbb{Q}$, and, for sufficiently large $N$, an effective version of Bogomolov’s conjecture for $X_1\left(N\right)/\mathbb{Q}$. ...

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.

Let $f$, $g$ and $h$ be three distinct primitive holomorphic cusp forms of even integral weights $k_1$, $k_2$ and $k_3$ for the full modular group $\Gamma =\mathrm{SL}(2,\mathbb{Z})$, respectively, and let ${\lambda }_f\left(n\right)$, ${\lambda }_g\left(n\right)$ and ${\lambda }_h\left(n\right)$ denote the $n$th normalized Fourier coefficients of $f$, $g$ and $h$, respectively. We consider the cancellations of sums related to arithmetic functions ${\lambda }_g\left(n\right)$, ${\lambda }_h\left(n\right)$ twisted by ${\lambda }_f\left(n\right)$ and establish the following results: $$\sum _{n\le x}{\lambda }_f\left(n\right){\lambda }_g{\left(n\right)}^i{\lambda }_h{\left(n\right)}^j{\ll }_{f,g,h,\epsilon }{x}^{1-1/2^{i+j}+\epsilon }$$ for any $\epsilon >0$, where $1\le i\le 2$, $j\ge 5$ are any fixed positive integers.

Following G. Grätzer and E. Knapp (2007), a slim planar semimodular lattice, SPS lattice for short, is a finite planar semimodular lattice having no $M_3$ as a sublattice. An SPS lattice is a slim rectangular lattice if it has exactly two doubly irreducible elements and these two elements are complements of each other. A finite poset $P$ is said to be JConSPS-representable if there is an SPS lattice $L$ such that $P$ is isomorphic to the poset $\mathrm{J}\left(\mathrm{Con}L\right)$ of join-irreducible congruences of $L$. We prove that...

We lift the notion of quasicontinuous posets to the topology context, called quasicontinuous spaces, and further study such spaces. The main results are: (1) A $T_0$ space $(X,\tau )$ is a quasicontinuous space if and only if $SI\left(X\right)$ is locally hypercompact if and only if $({\tau }_{SI},\subseteq )$ is a hypercontinuous lattice; (2) a $T_0$ space $X$ is an $SI$-continuous space if and only if $X$ is a meet continuous and quasicontinuous space; (3) if a $C$-space $X$ is a well-filtered poset under its specialization order, then $X$ is a quasicontinuous...