
Pentti HaukkanenTampere University | UTA · School of Information Sciences
Pentti Haukkanen
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Introduction
Publications
Publications (190)
This paper gives expressions for the solution $\{a(n)\}$ of the equation \begin{equation*} \sum_{k=0}^n{n \choose k}a(k)b(n-k)=c(n), \ n=0,1,2,\ldots, \end{equation*} where $b(0)\ne 0$, that is, of the equation $a \circ b = c$ in $a$, where $\circ$ is the binomial convolution. These expressions are classified as recursive, explicit, determinant, ex...
We study the solvability of the congruence x n "´a n pmod mq, where n, m P Z`, a P Z, and gcd pa, mq " 1. Our motivation arises from computer experiments concerning a geometric property of the roots of the congruence x n`yn " 0 pmod pq, where n P Z`and p P P. We encounter several OEIS sequences. We also make new observations on some of them.
A arithmetical function f is said to be a totient if there exist completely multiplicative functions ft\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_t$$\end{documen...
A positive integer is said to be an exponential divisor or an e-divisor of if 𝑑 𝑖 ∣ 𝑛 𝑖 for all prime divisors 𝑝 𝑖 of 𝑛. In addition, 1 is an e-divisor of 1. It is easy to see that ℤ + is a poset under the e-divisibility relation. Utilizing this observation we show that e-convolution of arithmetical functions is an example of the convolution of inc...
An arithmetical function f is multiplicative if f(1)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(1)=1$$\end{document} and f(mn)=f(m)f(n)\documentclass[12pt]{mini...
A perpendicularity ? in a module M is a binary relation that is irre ‡exive (but 0 ? 0), symmetric, serial, and preserves addition and scalar multiplication. If ? is not a subset of another perpendicularity in M , then ? is maximal. If M is a …nite-dimensional vector space and ? is induced by an inner product, then it is well known that ? is maxima...
We study existence of a solution of the arithmetical equation $f\ast h = g$ in $f,$ where $f\ast h$ is the Dirichlet convolution of arithmetical functions $f$ and $h,$ and derive an explicit expression for the solution. As applications we obtain expressions for the Möbius function $\mu$ and the so-called totients. As applications we also present ou...
We report from a Nordic research project that has investigated first-year engineering students in Norway, Sweden, and Finland, and the relationships between their task performance, motivational values, and beliefs about the nature of mathematics. The present paper focuses on the covariance between their motivational values and beliefs from the pers...
In 1876 H. J. S. Smith defined an LCM matrix as follows: let S = {x_1, x_2, ..., x_n} be a set of positive integers. The LCM matrix [S] is the n $\times$ n matrix with lcm(x_i , x_j) as its ij entry. During the last 30 years singularity of LCM matrices has interested many authors. In 1992 Bourque and Ligh ended up conjecturing that if the GCD close...
Two subsets P and Q of the set of positive integers is said to form a conjugate pair if each positive integer n possesses a unique factorization of the form n = ab, a ∈ P, b ∈ Q. In this paper we generalize conjugate pairs of sets to the setting of regular convolutions and study associated arithmetical functions. Particular attention is paid to ari...
A divisor d of a positive integer n is called a unitary divisor if \gcd(d, n/d)=1; and d is called a bi-unitary divisor of n if the greatest common unitary divisor of d and n/d is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let \sigma^{**}(n) denote the sum of the bi-unitary divisors of n. A positive integer n is ca...
We introduce a measure of dimensionality of an Abelian group. Our definition of dimension is based on studying perpendicularity relations in an Abelian group. For G ≅ ℤn, dimension and rank coincide but in general they are different. For example, we show that dimension is sensitive to the overall dimensional structure of a finite or finitely genera...
A divisor $d$ of a positive integer $n$ is called a unitary divisor if $\gcd(d, n/d)=1;$ and $d$ is called a bi-unitary divisor of $n$ if the greatest common unitary divisor of $d$ and $n/d$ is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let $\sig^{**}(n)$ denote the sum of the bi-unitary divisors of $n$. A positive...
Completely additive (c-additive in short) functions and completely multiplicative (c-multiplicative in short) functions are ordinarily defined for positive integers but sometimes on larger domains. We survey this matter by extending these functions first to nonzero integers and thereafter to nonzero rationals. Then we can similarly extend Leibniz-a...
A divisor d of a positive integer n is called a unitary divisor if \gcd(d, n/d)=1; and d is called a bi-unitary divisor of n if the greatest common unitary divisor of d and n/d is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let \sig^{**}(n) denote the sum of the bi-unitary divisors of n. A positive integer n is call...
Tässä artikkelissa tarkastellaan Tampereen yliopistossa DI-koulutuksessa aloittaneiden opiskelijoiden (N=272) matematiikkaan liittyvää motivaatiota, käsityksiä oppiaineen luonteesta ja menestystä lukiotasoisissa matematiikan tehtävissä. Kiinnostus matematiikkaa kohtaan ja hyvä käsitys itsestä matematiikan oppijana ovat vahvimpia osaamisen selittäji...
This study investigates Finnish, Norwegian, and Swedish first-year engineering students' task performance in mathematics and examines how it relates to their motivational values and beliefs about the nature of mathematics. In a set of seven mathematical tasks, female students outperformed male students, for example, in the simplification of symboli...
A divisor d of a positive integer n is called a unitary divisor if \gcd(d, n/d)=1; and d is called a bi-unitary divisor of n if the greatest common unitary divisor of d and n/d is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let \sigma^{**}(n) denote the sum of the bi-unitary divisors of n. A positive integer n is ca...
In a previous paper, we proved that the arithmetic subderivative D S is discontinuous at any rational point with respect to the ordinary absolute value. In the present paper, we study this question with respect to the p-adic absolute value. In particular, we show that DS is in this sense continuous at the origin if S is finite or p is not in S.
We give common generalizations of the Menon-type identities by Sivaramakrishnan (1969) and Li, Kim, Qiao (2019). Our general identities involve arithmetic functions of several variables, and also contain, as special cases, identities for gcd-sum type functions. We point out a new Menon-type identity concerning the lcm function.
We present a simple...
Let S={x1,x2,…,xn} be a finite set of distinct positive integers. Throughout this article we assume that the set S is GCD closed. The LCM matrix [S] of the set S is defined to be the n×n matrix with lcm(xi,xj) as its ij element. The famous Bourque-Ligh conjecture used to state that the LCM matrix of a GCD closed set S is always invertible, but curr...
The Kesava Menon norm of an arithmetical function f is defined by \(N(f)(n) = (f*\lambda f)(n^2)\), where \(*\) denotes the Dirichlet convolution and \(\lambda \) denotes Liouville’s function. The mth power Kesava Menon norm of f is defined inductively by \(N^0(f) = f\), \(N^m(f)=N\big (N^{m-1}(f)\big )\), \(m=1,2,\ldots \) In this paper we prove t...
We give common generalizations of the Menon-type identities by Sivaramakrishnan (1969) and Li, Kim, Qiao (2019). Our general identities involve arithmetic functions of several variables, and also contain, as special cases, identities for gcd-sum type functions. We point out a new Menon-type identity concerning the lcm function. We present a simple...
We first prove that any arithmetic subderivative of a rational number defines a function that is everywhere discontinuous in a very strong sense. Second, we show that although the restriction of this function to the set of integers is continuous (in the relative topology), it is not Lipschitz continuous. Third, we see that its restriction to a suit...
In the theory of Fourier transform some functions are said to be positive definite based on the positive definiteness property of a certain class of matrices associated with these functions. In the present article we consider how to define a similar positive definiteness property for arithmetical functions, whose domain is not the set of real numbe...
We first introduce the arithmetic subderivative of a positive integer with respect to a non-empty set of primes. This notion generalizes the concepts of the arithmetic derivative and arithmetic partial derivative. More generally, we then define that an arithmetic function $f$ is Leibniz-additive if there is a nonzero-valued and completely multiplic...
Let $S=\{x_1,x_2,\ldots,x_n\}$ be a finite subset of distinct positive integers. Throughout this article we also assume that our set $S$ is GCD closed. The LCM matrix $[S]$ of the set $S$ is defined to be the $n\times n$ matrix with $\mathrm{lcm}(x_i,x_j)$ as its $ij$ element. The famous Bourque-Ligh conjecture used to state that the LCM matrix of...
Nagell's totient θ(n, r) counts the number of solutions of the congruence (*) n ≡ x + y (mod r) under the restriction (x, r) = (y, r) = 1. In this paper we evaluate the number θ(n, r, q) of solutions of the congruence (*) under the restriction (x, r) = (y, r) = q, where q|r, via Ramanathan's approach to class-division of integers (mod r).
We express the values of the Dirichlet inverse f −1 in terms of the values of f without using the values of f −1. We use a method based on representing f −1 * f = δ as a system of linear equations. Jagannathan has given many of the results of this paper without proof starting from the basic recurrence relation for the values of f −1 .
An arithmetical function f is said to be weakly multiplicative if f is not identically zero and f (np) = f (n)f (p) for all positive integers n and primes p with (n, p) = 1. Every multiplicative function is a weakly multiplicative function but the converse is not true. In this note we study basic properties of weakly multiplicative functions with r...
We introduce a notion of positive definiteness for functions $f\!:P\to\mathbb{R}$ defined on meet semilattices $(P,\preceq,\wedge)$. To support our analysis, we present a new $LDL^{\rm T}$ decomposition for meet matrices subject to the product order. In addition, we explore the consequences of the lattice-theoretic definition in terms of multivaria...
An arithmetic function $f$ is Leibniz-additive if there is a completely multiplicative function $h_f$, i.e., $h_f(1)=1$ and $h_f(mn)=h_f(m)h_f(n)$ for all positive integers $m$ and $n$, satisfying $$ f(mn)=f(m)h_f(n)+f(n)h_f(m) $$ for all positive integers $m$ and $n$. A motivation for the present study is the fact that Leibniz-additive functions a...
An arithmetic function $f$ is Leibniz-additive if there is a completely multiplicative function $h_f$, i.e., $h_f(1)=1$ and $h_f(mn)=h_f(m)h_f(n)$ for all positive integers $m$ and $n$, satisfying $$ f(mn)=f(m)h_f(n)+f(n)h_f(m) $$ for all positive integers $m$ and $n$. A motivation for the present study is the fact that Leibniz-additive functions a...
Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobenius norm of the r-circulant matrix corresponding to g0, . . . , gn−1. We also give three lower bounds (with equality conditions) for the spectral norm of this matrix. For this purpose, we present three ways to estimate the spectral norm from below in gen...
We present the group-theoretic structure of the classes of multiplicative and firmly multiplicative arithmetical functions of several variables under the Dirichlet convolution, and we give characterizations of these two classes in terms of a derivation of arithmetical functions.
Let a_1,…,a_m be such real numbers that can be expressed as a (finite) product of prime powers with rational exponents. Using arithmetic partial derivatives, we define the arithmetic Jacobian matrix J_a of the vector a=(a_1,…,a_m) analogously to the Jacobian matrix J_f of a vector function f. We introduce the concept of multiplicative independence...
Let S = {x_1, x_2, . . . , x_n} be a set of distinct positive integers, and let f be an arithmetical function. The GCD matrix (S_)f on S associated with f is defined as the n × n matrix having f evaluated at the greatest common divisor of x_i and x_j as its ij entry. The LCM matrix [S]_f is defined similarly. We consider inertia, positive definiten...
We define perpendicularity in an Abelian group G as a binary relation satisfying certain five axioms. Such a relation is maximal if it is not a subrelation of any other perpendicularity in G. A motivation for the study is that the poset (𝒫, ⊆) of all perpendicularities in G is a lattice if G has a unique maximal perpendicularity, and only a meet-se...
Let $S=\{x_1,x_2,\dots,x_n\}$ be a set of distinct positive integers, and let $f$ be an arithmetical function. The GCD matrix $(S)_f$ on $S$ associated with $f$ is defined as the $n\times n$ matrix having $f$ evaluated at the greatest common divisor of $x_i$ and $x_j$ as its $ij$ entry. The LCM matrix $[S]_f$ is defined similarly. We consider inert...
It is well known that Euler’s totient function \(\phi \) satisfies the arithmetical equation \( \phi (mn)\phi ((m, n))=\phi (m)\phi (n)(m, n) \) for all positive integers m and n, where (m, n) denotes the greatest common divisor of m and n. In this paper we consider this equation in a more general setting by characterizing the arithmetical function...
Let $a$, $b$, $p$, $q$ be integers and~$(h_n)$ defined by $h_0=a$, $h_1=b$, $h_n=ph_{n-1}+qh_{n-2}$, $n=2,3,\dots$. Complementing to certain previously known results, we study the spectral norm of the circulant matrix corresponding to $h_0,\dots,h_{n-1}$.
Kovič, and implicitly Ufnarovski and Åhlander, defined a notion of arithmetic par-tial derivative. We generalize the definition for rational numbers and study several arithmetic partial differential equations of the first and second order. For some equa- tions, we give a complete solution, and for others, we extend previously known results. For exa...
Let T = {z1, z2, . . . , zn} be a finite multiset of real numbers, where z1 ≤ z2 ≤ · · · ≤ zn. The purpose
of this article is to study the different properties of MIN and MAX matrices of the set T with min(zi , zj) and
max(zi , zj) as their ij entries, respectively.We are going to do this by interpreting these matrices as so-called
meet and join ma...
We present how the education of subject teachers is organized in mathematics, science and computer science in Tampere. It is based on the idea that both engineering students and students from mathematics and science may choose to become a subject teacher. Students are accepted either to the master’s degree program in Science and Engineering of Tamp...
The invertibility of LCM matrices and their Hadamard powers have been studied
a lot over the years by many authors. Bourque and Ligh conjectured in 1992 that
the LCM matrix $[S]=[[x_i, x_j]]$ on any GCD closed set $S=\{x_1, x_2, \ldots,
x_n\}$ is invertible, but in 1997 this was proven false. However, currently
there are many open conjectures conce...
The Bourque-Ligh conjecture states that if $S=\{x_1,x_2,\ldots,x_n\}$ is a
gcd-closed set of positive integers with distinct elements, then the LCM matrix
$[S]=[\hbox{lcm}(x_i,x_j)]$ is invertible. It is well known that this
conjecture holds for $n\leq7$ but does not generally hold for $n\geq8$. In this
paper we provide a lattice-theoretic explanat...
Define n × n tridiagonal matrices T and S as follows: All entries of the main diagonal of T are zero and those of the first super- and subdiagonal are one. The entries of the main diagonal of S are two except the (n, n) entry one, and those of the first super- and subdiagonal are minus one. Then, denoting by λ(·) the largest eigenvalue,
Using cert...
Let Gn be a regular n-gon with unit circumradius. Let the edges and diagonals of Gn be en1 <...< enm. We compute the coefficients of the polynomial (x - e2n1)...(x - e2nμ). They appear to form a well-known integer sequence, and we study certain related sequences, too.We also compute the coefficients of the polynomial (x - s2n1)...(x - s2nm), where...
In this paper, we investigate which aspects are overriding in the concept images of monotonicity of Finnish tertiary mathematics students, i.e., on which aspects of monotonicity they base their argument in different types of exercises related to that concept. Further, we examine the relationship between the quality of principal aspects and the succ...
In this paper we study the structure and give bounds for the eigenvalues of
the $n\times n$ matrix, which $ij$ entry is $(i,j)^\alpha[i,j]^\beta$, where
$\alpha,\beta\in\Rset$, $(i,j)$ is the greatest common divisor of $i$ and $j$
and $[i,j]$ is the least common multiple of $i$ and $j$. Currently only
$O$-estimates for the greatest eigenvalue of th...
We give a set of axioms to establish a perpendicularity relation in an Abelian group and
then study the existence of perpendicularities in and
and in certain other groups. Our approach provides a justification for the use of the symbol
denoting relative primeness in number theory and extends the domain of this
convention to some degree. Related t...
Given n ∈ ℤ, its arithmetic derivative n' is defined as follows: (i) 0'=1'=(-1)'=0. (ii) If n = up1 · · · pk, where u = ±1 and p1,..., pk are primes (some of them possibly equal), then An analogous definition can be given in any unique factorization domain. What about the converse? Can the arithmetic derivative be (well-)defined on a non-unique fac...
An arithmetical function $f$ is said to be even (mod r) if f(n)=f((n,r)) for
all n\in\Z^+, where (n, r) is the greatest common divisor of n and r. We adopt
a linear algebraic approach to show that the Discrete Fourier Transform of an
even function (mod r) can be written in terms of Ramanujan's sum and may thus
be referred to as the Discrete Ramanuj...
In this paper we study the positive definiteness of meet and join matrices
using a novel approach. When the set $S_n$ is meet closed, we give a sufficient
and necessary condition for the positive definiteness of the matrix $(S_n)_f$.
From this condition we obtain some sufficient conditions for positive
definiteness as corollaries. We also use graph...
An integer a is said to be regular (modr) if there exists an integer x such that a
2x≡a (mod r). In this paper we introduce an analogue of Ramanujan’s sum with respect to regular integers (modr) and show that this analogue possesses properties similar to those of the usual Ramanujan’s sum.
In 1997, the first author [Nieuw Arch. Wiskd., IV. Ser. 15, No. 1-2, 73–77 (1997; Zbl 0928.11005)] showed that, if f is a completely multiplicative arithmetic function, then f α =μ -α f for any real number α, where μ α is the Souriau-Hsu-Möbius function. In 2002, V. Laohakosol, N. Pabhapote and N. Wechwiriyakul [Int. J. Math. Math. Sci. 29, No. 11,...
Considering lower closed sets as closed sets on a preposet (P, ≤), we obtain an Alexandroff topology on P. Then order preserving functions are continuous functions. In this article we investigate order preserving properties (and thus continuity properties) of integer-valued arithmetical functions under the usual divisibility relation of integers an...
We consider the classes of quasimultiplicative, semimultiplicative and
Selberg multiplicative functions as extensions of the class of multiplicative
functions. We apply these concepts to Ramanujan's sum and its analogue with
respect to regular integers (mod r).
We present an asymptotic formula for the number of line segments connecting
q+1 points of an nxn square grid, and a sharper formula, assuming the Riemann
hypothesis. We also present asymptotic formulas for the number of lines through
at least q points and, respectively, through exactly q points of the grid. The
well-known case q=2 is so generalized...
Let $(P,\preceq)$ be a lattice and $f$ a complex-valued function on $P$. We
define meet and join matrices on two arbitrary subsets $X$ and $Y$ of $P$ by
$(X,Y)_f=(f(x_i\wedge y_j))$ and $[X,Y]_f=(f(x_i\vee x_j))$ respectively. Here
we present expressions for the determinant and the inverse of $[X,Y]_f$. Our
main goal is to cover the case when $f$ i...
Let $(P,\preceq)$ be a lattice, $S$ a finite subset of $P$ and
$f_1,f_2,...,f_n$ complex-valued functions on $P$. We define row-adjusted meet
and join matrices on $S$ by $(S)_{f_1,...,f_n}=(f_i(x_i\wedge x_j))$ and
$[S]_{f_1,...,f_n}=(f_i(x_i\vee x_j))$. In this paper we determine the
structure of the matrix $(S)_{f_1,...,f_n}$ in general case and...
Let $m,n\ge 2$, $m\le n$. It is well-known that the number of
(two-dimensional) threshold functions on an $m\times n$ rectangular grid is
{eqnarray*} t(m,n)=\frac{6}{\pi^2}(mn)^2+O(m^2n\log{n})+O(mn^2\log{\log{n}})=
\frac{6}{\pi^2}(mn)^2+O(mn^2\log{m}). {eqnarray*} We improve the error term by
showing that $$ t(m,n)=\frac{6}{\pi^2}(mn)^2+O(mn^2). $...
Descartes’ rule of signs yields an upper bound for the number of positive and negative real roots of a given polynomial. The fundamental theorem of algebra implies a similar property; every real polynomial of degree n⩾1 has at most n real zeroes. In this paper, we describe axiomatically function families possessing one or another of these propertie...
Let l(n) be the number of lines through at least two points of an n × n rectangular grid. We prove recursive and asymptotic formulas for it using respectively combinatorial and number theoretic
methods. We also study the ratio l(n)/l(n − 1). All this originates from Mustonen’s experimental results.
We present a formula for the number of line segments connecting q+1 points of
an n_1 x...x n_k rectangular grid. As corollaries, we obtain formulas for the
number of lines through at least q points and, respectively, through exactly q
points of the grid. The well-known case k=2 is so generalized. We also present
recursive formulas for these numbers...
In 1861, Henry John Stephen Smith [H.J.S. Smith, On systems of linear indeterminate equations and congruences, Philos. Trans. Royal Soc. London. 151 (1861), pp. 293–326] published famous results concerning solving systems of linear equations. The research on Smith normal form and its applications started and continues. In 1876, Smith [H.J.S. Smith,...
We give a detailed study of the discrete Fourier transform (DFT) of r-even arithmetic functions, which form a subspace of the space of r-periodic arithmetic functions. We consider the DFT of sequences of r-even functions, their mean values and Dirichlet series. Our results generalize properties of the Ramanujan sum. We show that some known properti...
We establish a framework in which one can study perpendicularity and parallelism axiomatically. The lines of the Euclidean plane provide an admissible model of our axiom system but various other models exist, too. Focusing only on these two relations, our approach is more elementary and, thus, more suitable for the teaching of deductive geometry in...
The fundamental theorem of algebra implies that every real polynomial of degree n≥1 has at most n real zeros. Descartes’ rule of signs determines the maximum number of positive and negative real roots of a polynomial. We propose function families possessing these properties on the number of real zeros. These function families include polynomial fun...
An integer $a$ is said to be regular (mod $r$) if there exists an integer $x$ such that $a^2x\equiv a\pmod{r}$. In this paper we introduce an analogue of Ramanujan's sum with respect to regular integers (mod $r$) and show that this analogue possesses properties similar to those of the usual Ramanujan's sum.
A divisor d + of n + is said to be a unitary divisor of n if (d, n/d) = 1. In this article we examine the greatest common unitary divisor (GCUD) reciprocal least common unitary multiple (LCUM) matrices. At first we concentrate on the difficulty of the non-existence of the LCUM and we present three different ways to overcome this difficulty. After t...
We show that there are relationships between a generalized Lucas sequence and the permanent and determinant of some Hessenberg matrices.
Let h(x) be a polynomial with real coefficients. We introduce h(x)-Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h(x)-Fibonacci polynomials. We also introduce h(x)-Lucas polynomials that generalize the Lucas polynomial...
Let $n=\prod_p p^{\nu_p(n)}$ denote the canonical factorization of $n\in \N$. The binomial convolution of arithmetical functions $f$ and $g$ is defined as $(f\circ g)(n)=\sum_{d\mid n} (\prod_p \binom{\nu_p(n)}{\nu_p(d)}) f(d)g(n/d),$ where $\binom{a}{b}$ is the binomial coefficient. We provide properties of the binomial convolution. We study the $...
It is well-known that (ℤ+, |) = (ℤ+, GCD, LCM) is a lattice, where | is the usual divisibility relation and GCD and LCM stand for the greatest common divisor
and the least common multiple of positive integers.
The number $
d = \prod\nolimits_{k = 1}^r {p_k^{d^{(k)} } }
$
d = \prod\nolimits_{k = 1}^r {p_k^{d^{(k)} } }
is said to be an exponential...
Let (P,⩽)=(P,∧,∨) be a lattice, let S={x1,x2,…,xn} be a meet-closed subset of P and let f:P→Z+ be a function. We characterize the matrix divisibility of the join matrix [S]f=[f(xi∨xj)] by the meet matrix (S)f=[f(xi∧xj)] in the ring Zn×n in terms of the usual divisibility in Z, and we present two algorithms for constructing certain classes of meet-c...