Determinant and Inverse of Meet and Join Matrices

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Authors: Altinisik Ercan, Tuglu Naim, Haukkanen Pentti Name of article: Determinant and Inverse of Meet and Join Matrices Year of publication: 2007

Name of journal: International Journal of Mathematics and Mathematical Sciences

Volume: 2007

Number of issue: 37580

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ISSN: 1687-0425

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Volume 2007, Article ID 37580,11pages doi:10.1155/2007/37580

Research Article

Determinant and Inverse of Meet and Join Matrices

Ercan Altinisik, Naim Tuglu, and Pentti Haukkanen

Received 2 March 2007; Accepted 18 April 2007 Recommended by Dihua Jiang

We define meet and join matrices on two subsetsX andY of a lattice (P,) with respect to a complex-valued function f on P by (X,Y)_f =(f(x_i∧y_i)) and [X,Y]_f = (f(xi∨yi)), respectively. We present expressions for the determinant and the inverse of (X,Y)_f and [X,Y]_f, and as special cases we obtain several new and known formulas for the determinant and the inverse of the usual meet and join matrices (S)_f and [S]_f. Copyright © 2007 Ercan Altinisik et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetS= {x1,x2,...,xn}be a set of distinct positive integers, and let f be an arithmetical function. Let (S)f denote then×nmatrix having f evaluated at the greatest common divisor (x_i,x_j) ofx_iandx_jas itsi j-entry, that is, (S)_f =(f((x_i,x_j))). Analogously, let [S]_f denote then×nmatrix having f evaluated at the least common multiple [x_i,x_j] ofx_i andxjas itsi j-entry, that is, [S]f =(f([xi,xj])). The matrices (S)f an [S]f are referred to as the GCD and LCM matrices onSassociated with f, respectively. The setSis said to be factor-closed if it contains every divisor ofxfor anyx∈S, and the setSis said to be GCD-closed if (xi,xj)∈Swheneverxi,xj∈S. Every factor-closed set is GCD-closed but the converse does not hold.

Smith [1] calculated det(S)_f whenSis factor-closed and det[S]_f in a more special case.

Since Smith, a large number of results on GCD and LCM matrices have been presented in the literature. For general accounts, see, for example, [2–5].

Let (P,) be a lattice in which every principal order ideal is finite. LetS={x1,x2,...,x_n} be a subset ofP, and let f be a complex-valued function onP. Then then×nmatrix (S)f =(f(xi∧xj)) is called the meet matrix onSassociated with f and then×nmatrix

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[S]f =(f(xi∨xj)) is called the join matrix onSassociated withf. If (P,)=(Z⁺,|), then meet and join matrices, respectively, become GCD and LCM matrices. The setSis said to be lower-closed (resp., upper-closed) if for everyx,y∈Pwithx∈Sandyx(resp., xy), we havey∈S. The setSis said to be meet-closed (resp., join-closed) if for every x,y∈S, we havex∧y∈S(resp.,x∨y∈S).

Meet matrices have been studied in many papers (see, e.g., [3,5–11]). Haukkanen [7]

calculated the determinant of (S)f on arbitrary setSand obtained the inverse of (S)f

on a lower-closed setS. Korkee and Haukkanen [12] obtained the inverse of (S)f on a meet-closed setS.

Join matrices have previously been studied by Hong and Sun [13], Korkee and Haukka- nen [5], and Wang [11]. Korkee and Haukkanen [5] present, among others, formulas for the determinant and inverse of [S]_f on meet-closed, join-closed, lower-closed, and upper-closed setsS.

LetX= {x1,x2,...,xn}andY= {y1,y2,...,yn}be two subsets ofP. We define the meet matrix onX andY with respect to f as (X,Y)_f =(f(x_i∧y_j)). In particular, (S,S)_f = (S)_f. Analogously, we define the join matrix onXandY with respect to f as [X,Y]_f = (f(xi∨yj)). In particular, [S,S]f=[S]f.

In this paper we present expressions for the determinant and the inverse of (X,Y)_f on arbitrary setsXandY. IfX=Y =S, then we obtain the determinant formula for (S)_f given in [7] and a formula for the inverse of (S)f on arbitrary setS. IfSis meet-closed or lower-closed, then the formula for the inverse of (S)f reduces to those given in [7,12].

We also obtain a new expression for the inverse formulas of (S)_f given in [7,12].

We also present expressions for the determinant and inverse of [X,Y]_f when the function f is semimultiplicative (for definition, see (6.1)). As special cases, we obtain formulas for the determinant and inverse of [S]_f on arbitrary setS. These formulas generalize the determinant and the inverse formulas of [S]f on meet-closed and lower-closed setsS presented in [5]. As special cases, we also obtain some new and known results on LCM matrices.

Determinant and inverse formulas for (S)f and [S]f on join-closed and upper-closed setsScould be obtained applying duality to the results of this paper. We do not include the details of these results here.

2. Preliminaries

Let (P,) be a lattice in which every principal order ideal is finite, and let f be a complex- valued function onP. LetX= {x1,x2,...,xn}andY= {y1,y2,...,yn}be two subsets ofP.

Let the elements ofXandY be arranged so thatxixj⇒i≤jand yiyj⇒i≤j. Let D= {d1,d2,...,d_m}be any subset ofPcontaining the elementsx_i∧y_j,i,j=1, 2,...,n. Let the elements ofDbe arranged so thatdidj⇒i≤j. We define the functionΨD,f onD inductively as

ΨD,f

d_k=fd_k−

dv≺dk

ΨD,f

d_v (2.1)

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or

fdk

=

dvdk

ΨD,f

dv

. (2.2)

Then

ΨD,f dk

=

dvdk

fdv μD

dv,dk

, (2.3)

whereμ_Dis the M¨obius function of the poset (D,), see [14, Section IV.1]. IfDis meet- closed, then

ΨD,f dk

=

zdk

z dt

t<k

wz

f(w)μ(w,z), (2.4)

whereμis the M¨obius function ofP, and ifDis lower-closed, then ΨD,f

d_k=

dvdk

fd_vμd_v,d_k, (2.5) whereμis the M¨obius function ofP. For proofs of (2.4) and (2.5), see [7]. If (P,)= (Z⁺,|) andDis factor-closed, thenμD(dv,dk)=μ(dk/dv) (see [15, Chapter 7]), whereμis the number-theoretic M¨obius function, and (2.3) becomes

ΨD,f dk

=

dv|dk

fdv μ

d_k dv

=

f ∗μdk

, (2.6)

where∗is the Dirichlet convolution of arithmetical functions.

LetE(X)=(e_{i j}(X)) andE(Y)=(e_{i j}(Y)) denote then×mmatrices defined by e_{i j}(X)=

⎧⎨

⎩

1 ifd_jx_i, 0 otherwise, e_{i j}(Y)=

⎧⎨

⎩

1 ifd_jy_i, 0 otherwise,

(2.7)

respectively. Note thatE(X) andE(Y) depend onDbut for the sake of brevity,Dis omit- ted from the notation. We also denote

ΛD,f =diagΨD,f

d1

,ΨD,f

d2

,...,ΨD,f

d_m. (2.8)

3. A structure theorem

In this section, we give a factorization of the matrix (X,Y)_f =(f(xi∧yj)). As special cases, we obtain the factorizations of (S)_f given in [7,9,12]. A large number of similar factorizations are presented in the literature. The idea of this kind of factorization may be considered to originate from P ´olya and Szeg¨o [16].

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Theorem 3.1. One has

(X,Y)_f =E(X)ΛD,fE(Y)^T. (3.1) Proof. By (2.2), thei j-entry of (X,Y)_f is

fx_i∧y_j=

dvxi∧yj

ΨD,f

d_v. (3.2)

Now, applying (2.7) and (2.8) to (3.2), we obtainTheorem 3.1.

Remark 3.2. The setsXandYcould be allowed to have distinct cardinalities in Theorems 3.1and6.1. However, in other results we must assume that these cardinalities coincide.

4. Determinant formulas

In this section, we derive formulas for determinants of meet matrices. InTheorem 4.1, we present an expression for det(X,Y)_f on arbitrary setsXandY. TakingX=Y=S= {x1,x2,...,x_n}inTheorem 4.1, we could obtain a formula for the determinant of usual meet matrices (S)f on arbitrary setS(see [7, Theorem 3]), and further taking (P,)= (Z⁺,|) we could obtain a formula for the determinant of GCD matrices on arbitrary set S(see [17, Theorem 2]). In Theorems4.2and4.4, respectively, we calculate det(S)_f on meet-closed and lower-closed setsS. These formulas are also given in [7].

Theorem 4.1. (i) Ifn > m, then det(X,Y)_f =0.

(ii) Ifn≤m, then

det(X,Y)_f =

1_≤k1<k2<···<kn≤m

detE(X)(k1,k2,...,kn)detE(Y)(k1,k2,...,kn)

×ΨD,f

d_k1

ΨD,f

d_k2

···ΨD,f

d_kn

.

(4.1)

Proof. ByTheorem 3.1,

det(X,Y)_f =detE(X)ΛD,fE(Y)^T. (4.2)

Thus by the Cauchy-Binet formula, we obtainTheorem 4.1.

Theorem 4.2. IfSis meet-closed, then det(S)_f =ⁿ

v=1

ΨS,f

x_v=ⁿ

v=1

zxv

z xt

t<v

wz

f(w)μ(w,z). (4.3)

Proof. We takeX=Y =SinTheorem 4.1. SinceSis meet-closed, we may further take D=S. Thenm=nand detE(S)(k1,k2,...,kn)=detE(S)(1,2,...,n)=1, and so we obtain the first equality in (4.3). The second equality follows from (2.4).

Remark 4.3. Theorem 4.2can also be proved by takingX=Y=SandD=SinTheorem 3.1.

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Theorem 4.4. IfSis lower-closed, then

det(S)_f =ⁿ

v=1

ΨS,f

x_v=ⁿ

v=1

xuxv

fx_uμx_u,x_v. (4.4)

Proof. The first equality in (4.4) follows from (4.3). The second equality follows from

(2.5).

Corollary 4.5 [18, Theorem 2]. LetSbe a GCD-closed set of distinct positive integers, and let f be an arithmetical function. Then

det(S)f =ⁿ

v=1

z|xv

zxt

t<v

(f∗μ)(z). (4.5)

Corollary 4.6 [1]. LetSbe a factor-closed set of distinct positive integers, and let f be an arithmetical function. Then

det(S)f =ⁿ

v=1

(f ∗μ)xv

. (4.6)

5. Inverse formulas

In this section, we derive formulas for inverses of meet matrices. In Theorem 5.1, we present an expression for the inverse of (X,Y)_f on arbitrary setsXandY, and inTheorem 5.2we present an expression for the inverse of (S)_f on arbitrary setS. Taking (P,)= (Z⁺,|), we could obtain a formula for the inverse of GCD matrices on arbitrary setS.

Formulas for the inverse of meet or GCD matrices on arbitrary set have not previously been presented in the literature. In Theorems5.3and5.5, respectively, we calculate the inverse of (S)f on meet-closed and lower-closed setsS. Similar formulas are given in [12, Theorem 7.1] and [7, Theorem 6].

Theorem 5.1. LetX_i=X\ {x_i}andY_i=Y\ {y_i}fori=1, 2,...,n. If (X,Y)_f is invertible, then the inverse of (X,Y)_f is then×nmatrixB=(bi j), where

bi j= (−1)ⁱ⁺^j det(X,Y)_f

1≤k¹<k²<···<kn−1≤m

detEXj

(k¹,k²,...,kn−1)detEYi

(k¹,k²,...,kn−1)

×ΨD,f dk1

ΨD,f dk2

···ΨD,f dkn−1

.

(5.1)

Proof. It is well known that

bi j= αji

det(X,Y)_f, (5.2)

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whereαjiis the cofactor of theji-entry of (X,Y)_f. It is easy to see thatαji=(−1)ⁱ⁺^jdet(Xj, Yi)_f. ByTheorem 4.1, we see that

detX_j,Y_i_f =

1_≤k1<k2<···<kn−1≤m

detEX_j₍_k₁_,_k₂_,_..._,_k_n₋₁₎detEY_i₍_k₁_,_k₂_,_..._,_k_n₋₁₎

×ΨD,f

d_k1

ΨD,f

d_k2

···ΨD,f

d_kn−1

.

(5.3)

Combining the above equations, we obtainTheorem 5.1.

Theorem 5.2. LetSi=S\ {xi}fori=1, 2,...,n. If (S)f is invertible, then the inverse of (S)f

is then×nmatrixB=(b_{i j}), where bi j=(−1)ⁱ⁺^j

det(S)f

1≤k¹<k²<···<kn−1≤m

detESj

(k¹,k²,...,kn−1)detESi

(k¹,k²,...,kn−1)

×ΨD,f dk1

ΨD,f dk2

···ΨD,f dkn−1

.

(5.4)

Proof. TakingX=Y=SinTheorem 5.1, we obtainTheorem 5.2.

Theorem 5.3. Suppose thatSis meet-closed. If (S)f is invertible, then the inverse of (S)f is then×nmatrixB=(bi j), where

bi j= n k=1

(−1)ⁱ⁺^j ΨS,f

xk

detES^k_idetES^k_j, (5.5)

whereE(S^k_i) is the (n−1)×(n−1) submatrix ofE(S) obtained by deleting theith row and thekth column ofE(S), or

b_{i j}=

xi∨xjxk

μ_Sx_i,x_kμ_Sx_j,x_k ΨS,f

x_k , (5.6)

whereμSis the M¨obius function of the poset (S,).

Proof. SinceS is meet-closed, we may take D=S. ThenE(S) is a square matrix with detE(S)=1. Further,E(S)^T is the matrix associated with the zeta function of the finite poset (S,). Thus the inverse ofE(S)^T is the matrix associated with the M¨obius function of (S,), that is, ifU=(ui j) is the inverse ofE(S)^T, thenui j=μS(xi,xj), see [14, Section IV.1]. On the other hand,ui j=βi j/detE(S)^T=βi j, whereβi jis the cofactor of thei j-entry ofE(S). Hereβ_{i j}=(−1)ⁱ⁺^jdetE(S_i^j). Thus

(−1)ⁱ⁺^jdetES_i^j=μS xi,xj

. (5.7)

Now we applyTheorem 5.2withD=S. Thenm=n, and using formulas(4.3) and (5.7),

we obtainTheorem 5.3.

Remark 5.4. Equation (5.6) is given in [7, Theorem 6] and can also be proved by taking X =Y =S and D=S in Theorem 3.1 and then applying the formula (S)⁻_f¹ = (E(S)^T)⁻¹Λ⁻_S,¹fE(S)⁻¹.

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Theorem 5.5. Suppose thatSis lower-closed. If (S)f is invertible, then the inverse of (S)f is then×nmatrixB=(bi j), where

bi j=

xi∨xjxk

μxi,xk

μxj,xk ΨS,f

xk . (5.8)

Hereμis the M¨obius function of (P,).

Proof. SinceSis lower-closed, we haveμS=μonS, (apply [14, Proposition 4.6]). Thus,

Theorem 5.5follows fromTheorem 5.3.

Corollary 5.6 [18, Corollary 1]. LetSbe a factor-closed set of distinct positive integers, and let f be an arithmetical function. If (S)_f is invertible, then the inverse of (S)_f is then×n matrixB=(bi j), where

bi j=

[xi,xj]_|xk

μx_k/x_iμx_k/x_j (f∗μ)xk

. (5.9)

Hereμis the number-theoretic M¨obius function.

6. Formulas for join matrices

Let f be a complex-valued function onP. We say that f is a semimultiplicative function if

f(x)f(y)=f(x∧y)f(x∨y) (6.1) for allx,y∈P(see [5]).

The notion of a semimultiplicative function arises from the theory of arithmetical functions. Namely, an arithmetical functionf is said to be semimultiplicative iff(r)f(s)= f((r,s))f([r,s]) for allr,s∈Z⁺. For semimultiplicative arithmetical functions, reference is made to the book by Sivaramakrishnan [19], see also [2]. Note that a semimultiplicative arithmetical function f with f(1) =0 is referred to as a quasimultiplicative arithmetical function. Quasimultiplicative arithmetical functions with f(1)=1 are the usual multiplicative arithmetical functions.

In this section, we show that join matrices [X,Y]f with respect to semimultiplicative functions f possess properties similar to those given for meet matrices (X,Y)_f with respect to arbitrary functions f in Sections3,4, and5. Throughout this section, f is a semimultiplicative function onPsuch that f(x) =0 for allx∈P.

Theorem 6.1. One has

[X,Y]_f=ΔX,f(X,Y)1/ fΔY,f (6.2) or

[X,Y]_f =ΔX,fE(X)ΛD,1/ fE(Y)^TΔY,f, (6.3)

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where

ΔX,f =diagfx1 ,fx2

,...,fxn

. (6.4)

Proof. By (6.1), thei j-entry of [X,Y]_f is fxi∨yj

=fxi 1

fxi∧yjfyj

. (6.5)

We thus obtain (6.2), and applyingTheorem 3.1we obtain (6.3).

From (6.2), we obtain

det[X,Y]_f= _n

v=1

fx_vfy_v

det(X,Y)1/ f, [X,Y]⁻_f¹=Δ⁻_Y¹,f(X,Y)⁻₁_{/ f}¹Δ⁻_X¹,f.

(6.6)

Now, using (6.6) and the formulas of Sections4and5, we obtain formulas for join matrices.

We first present formulas for the determinant of join matrices. InTheorem 6.2, we give an expression for det[X,Y]_f on arbitrary setsXandY. Formulas for the determinant of join or LCM matrices on arbitrary sets have not previously been presented in the literature. In Theorems6.3and6.4, respectively, we calculate det[S]_f on meet-closed and lower-closed setsS. Similar formulas are given in [5, Section 5.3].

Theorem 6.2. (i) Ifn > m, then det[X,Y]_f =0.

(ii) Ifn≤m, then det[X,Y]_f =

_n

v=1

fxv

fyv

1≤k¹<k²<···<kn≤m

detE(X)(k1,k2,...,kn)detE(Y)_(k1,k2,...,kn)

×ΨD,1/ f

d_k1

ΨD,1/ f

d_k2

···ΨD,1/ f

d_kn

.

(6.7) Theorem 6.3. ifSis meet-closed, then

det[S]f =ⁿ

v=1

fxv2

ψs,1/ f xv

=ⁿ

v=1

fxv2

zxv

z xt

t<v

wz

μ(w,z)

f(w) . (6.8)

Theorem 6.4. IfSis lower-closed, then det[S]f =ⁿ

v=1

fxv2

ΨS,1/ f xv

=ⁿ

v=1

fxv2

xuxv

μxu,xv

fxu . (6.9)

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Corollary 6.5 [2, Theorem 3.2]. LetSbe a GCD-closed set of distinct positive integers, and let f be a quasimultiplicative arithmetical function such that f(r) =0 for allr∈Z⁺. Then

det[S]_f =ⁿ

v=1

fx_v²

z|xv

zxt

t<v

1 f ^∗μ

(z). (6.10)

Corollary 6.6 [20, Theorem 2]. LetSbe a factor-closed set of distinct positive integers, and let f be a quasimultiplicative arithmetical function such that f(r) =0 for allr∈Z⁺. Then

det[S]f =ⁿ

v=1

fxv2 1

f ^∗μxv

. (6.11)

We next derive formulas for inverses of join matrices. InTheorem 6.7, we give an expression for the inverse of [X,Y]_f on arbitrary setsXandY, and inTheorem 6.8we give an expression for the inverse of [S]f on arbitrary setS. Taking (P,)=(Z⁺,|) we could obtain a formula for the inverse of LCM matrices on arbitrary setS. Formulas for the inverse of join or LCM matrices on arbitrary set have not previously been presented in the literature. In Theorems6.9and6.10, respectively, we calculate the inverse of [S]f on meet-closed and lower-closed setsS. Similar formulas are given in [5, Section 5.3].

Theorem 6.7. LetX_i=X\ {x_i}andY_i=Y\ {y_i}fori=1, 2,...,n. If [X,Y]_f is invertible, then the inverse of [X,Y]_f is then×nmatrixB=(b_{i j}), where

b_{i j}= (−1)ⁱ⁺^j fx_jfy_idet[X,Y]_f

_n

v=1

fx_vfy_v

×

1≤k¹<k²<···<kn−1≤m

detEXj

(k¹,k²,...,kn−1)detEYi

(k¹,k²,...,kn−1)

×ΨD,1/ f dk1

ΨD,1/ f dk2

···ΨD,1/ f dkn−1

.

(6.12)

Theorem 6.8. LetSi=S\ {xi}fori=1, 2,...,n. If [S]f is invertible, then the inverse of [S]f

is then×nmatrixB=(b_{i j}), where bi j= (−1)ⁱ⁺^j

fxi

fxj

det[S]f

_n

v=1

fxv

2

×

1_≤k1<k2<···<kn−1≤m

detES_i₍_k₁_,_k₂_,_..._,_k_n₋₁₎detES_j₍_k₁_,_k₂_,_..._,_k_n₋₁₎

×ΨD,1/ f

d_k1

ΨD,1/ f

d_k2

···ΨD,1/ f

d_kn−1

.

(6.13)

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Theorem 6.9. Suppose thatSis meet-closed. If [S]f is invertible, then the inverse of [S]f is then×nmatrixB=(bi j), where

b_{i j}= 1 fx_ifx_j

xi∨xjxk

μ_Sx_i,x_kμ_Sx_j,x_k ΨS,1/ f

x_k . (6.14)

HereμSis the M¨obius function of the poset (S,).

Theorem 6.10. Suppose thatSis lower-closed. If [S]f is invertible, then the inverse of [S]f

is then×nmatrixB=(bi j), where b_{i j}= 1

fx_ifx_j

xi∨xjxk

μx_i,x_kμx_j,x_k ΨS,1/ f

x_k . (6.15)

Hereμis the M¨obius function of (P,).

Corollary 6.11 [20, Theorem 2]. LetSbe a factor-closed set of distinct positive integers, and let f be a quasimultiplicative arithmetical function such that f(r) =0 for allr∈Z⁺. If [S]_f is invertible, then the inverse of [S]_f is then×nmatrixB=(b_{i j}), where

bi j= 1 fxi

fxj

[xi,xj]|xk

μxk/xi

μxk/xj

(1/ f)∗μxk. (6.16)

Hereμis the number-theoretic M¨obius function.

References

[1] H. J. S. Smith, “On the value of a certain arithmetical determinant,” Proceedings of the London Mathematical Society, vol. 7, no. 1, pp. 208–213, 1875.

[2] P. Haukkanen and J. Sillanp¨a¨a, “Some analogues of Smith’s determinant,” Linear and Multilinear Algebra, vol. 41, no. 3, pp. 233–244, 1996.

[3] P. Haukkanen, J. Wang, and J. Sillanp¨a¨a, “On Smith’s determinant,” Linear Algebra and Its Ap- plications, vol. 258, pp. 251–269, 1997.

[4] S. Hong, “GCD-closed sets and determinants of matrices associated with arithmetical func- tions,” Acta Arithmetica, vol. 101, no. 4, pp. 321–332, 2002.

[5] I. Korkee and P. Haukkanen, “On meet and join matrices associated with incidence functions,”

Linear Algebra and Its Applications, vol. 372, pp. 127–153, 2003.

[6] E. Altinisik, B. E. Sagan, and N. Tuglu, “GCD matrices, posets, and nonintersecting paths,” Lin- ear and Multilinear Algebra, vol. 53, no. 2, pp. 75–84, 2005.

[7] P. Haukkanen, “On meet matrices on posets,” Linear Algebra and Its Applications, vol. 249, no. 1–

3, pp. 111–123, 1996.

[8] I. Korkee and P. Haukkanen, “On a general form of meet matrices associated with incidence functions,” Linear and Multilinear Algebra, vol. 53, no. 5, pp. 309–321, 2005.

[9] B. V. Rajarama Bhat, “On greatest common divisor matrices and their applications,” Linear Al- gebra and Its Applications, vol. 158, pp. 77–97, 1991.

[10] J. S´andor and B. Crstici, Handbook of Number Theory. II, Kluwer Academic Publishers, Dor- drecht, The Netherlands, 2004.

[11] B. Y. Wang, “Explicit expressions of Smith’s determinant on a poset,” Acta Mathematica Sinica, vol. 17, no. 1, pp. 161–168, 2001.

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[12] I. Korkee and P. Haukkanen, “Bounds for determinants of meet matrices associated with inci- dence functions,” Linear Algebra and Its Applications, vol. 329, no. 1–3, pp. 77–88, 2001.

[13] S. Hong and Q. Sun, “Determinants of matrices associated with incidence functions on posets,”

Czechoslovak Mathematical Journal, vol. 54, no. 2, pp. 431–443, 2004.

[14] M. Aigner, Combinatorial Theory, vol. 234 of Fundamental Principles of Mathematical Sciences, Springer, New York, NY, USA, 1979.

[15] P. J. McCarthy, Introduction to Arithmetical Functions, Universitext, Springer, New York, NY, USA, 1986.

[16] G. P ólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis. Band II: Funktionentheorie, Nullstellen, Polynome Determinanten, Zahlentheorie. Vierte Auflage, Springer, Berlin, Germany, 1971.

[17] Z. Li, “The determinants of GCD matrices,” Linear Algebra and Its Applications, vol. 134, pp.

137–143, 1990.

[18] K. Bourque and S. Ligh, “Matrices associated with arithmetical functions,” Linear and Multilin- ear Algebra, vol. 34, no. 3-4, pp. 261–267, 1993.

[19] R. Sivaramakrishnan, Classical Theory of Arithmetic Functions, vol. 126 of Monographs and Text- books in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1989.

[20] K. Bourque and S. Ligh, “Matrices associated with multiplicative functions,” Linear Algebra and Its Applications, vol. 216, pp. 267–275, 1995.

Ercan Altinisik: Department of Mathematics, Faculty of Arts and Sciences, Gazi University, 06500 Teknikokullar, Ankara, Turkey

Email address:ealtinisik@gazi.edu.tr

Naim Tuglu: Department of Mathematics, Faculty of Arts and Sciences, Gazi University, 06500 Teknikokullar, Ankara, Turkey

Email address:naimtuglu@gazi.edu.tr

Pentti Haukkanen: Department of Mathematics, Statistics and Philosophy, University of Tampere, 33014 Tampere, Finland

Email address:pentti.haukkanen@uta.fi