Determinant and Inverse of Meet and Join Matrices

We define meet and join matrices on two subsets X and Y of a lattice (P, ) with respect to a complex-valued function f on P by (X ,Y) f = ( f (xi ∧ yi)) 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 .


Introduction
Let S = {x 1 ,x 2 ,...,x n } be a set of distinct positive integers, and let f be an arithmetical function. Let (S) f denote the n × n matrix having f evaluated at the greatest common divisor (x i ,x j ) of x i and x j as its i j-entry, that is, (S) f = ( f ((x i ,x j ))). Analogously, let [S] f denote the n × n matrix having f evaluated at the least common multiple [x i ,x j ] of x i and x j as its i j-entry, that is, ). The matrices (S) f an [S] f are referred to as the GCD and LCM matrices on S associated with f , respectively. The set S is said to be factor-closed if it contains every divisor of x for any x ∈ S, and the set S is said to be GCD-closed if (x i ,x j ) ∈ S whenever x i ,x j ∈ S. Every factor-closed set is GCD-closed but the converse does not hold.
Smith [1] calculated det(S) f when S is 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][3][4][5].
Let (P, ) be a lattice in which every principal order ideal is finite. Let S={x 1 ,x 2 ,...,x n } be a subset of P, and let f be a complex-valued function on P. Then the n × n matrix (S) f = ( f (x i ∧ x j )) is called the meet matrix on S associated with f and the n × n matrix 2 International Journal of Mathematics and Mathematical Sciences [S] f = ( f (x i ∨ x j )) is called the join matrix on S associated with f . If (P, ) = (Z + ,|), then meet and join matrices, respectively, become GCD and LCM matrices. The set S is said to be lower-closed (resp., upper-closed) if for every x, y ∈ P with x ∈ S and y x (resp., x y), we have y ∈ S. The set S is said to be meet-closed (resp., join-closed) if for every x, y ∈ S, we have x ∧ y ∈ S (resp., x ∨ y ∈ S).
Join matrices have previously been studied by Hong and Sun [13], Korkee and Haukkanen [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 sets S.
Let X = {x 1 ,x 2 ,...,x n } and Y = {y 1 , y 2 ,..., y n } be two subsets of P. We define the meet matrix on X and Y 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 on X and Y with respect to f as In this paper we present expressions for the determinant and the inverse of (X,Y ) f on arbitrary sets X and Y . If X = 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 set S. If S is 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 set S. These formulas generalize the determinant and the inverse formulas of [S] f on meet-closed and lower-closed sets S 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 sets S could be obtained applying duality to the results of this paper. We do not include the details of these results here.

Preliminaries
Let (P, ) be a lattice in which every principal order ideal is finite, and let f be a complexvalued function on P. Let X = {x 1 ,x 2 ,...,x n } and Y = {y 1 , y 2 ,..., y n } be two subsets of P. Let the elements of X and Y be arranged so that x i x j ⇒ i ≤ j and y i y j ⇒ i ≤ j. Let D = {d 1 ,d 2 ,...,d m } be any subset of P containing the elements x i ∧ y j , i, j = 1,2,...,n. Let the elements of D be arranged so that d i d j ⇒ i ≤ j. We define the function Ψ D, f on D inductively as Then where μ D is the Möbius function of the poset (D, ), see [14,Section IV.1]. If D is meetclosed, then where μ is the Möbius function of P, and if D is lower-closed, then where μ is the Möbius function of P. For proofs of (2.4) and (2.5), see [7]. If (P, ) = (Z + ,|) and D is factor-closed, then [15,Chapter 7]), where μ is the number-theoretic Möbius function, and (2.3) becomes where * is the Dirichlet convolution of arithmetical functions. Let E(X) = (e i j (X)) and E(Y ) = (e i j (Y )) denote the n × m matrices defined by respectively. Note that E(X) and E(Y ) depend on D but for the sake of brevity, D is omitted from the notation. We also denote (2.8)

A structure theorem
In this section, we give a factorization of the matrix (X,Y ) f = ( f (x i ∧ y j )). 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 Pólya and Szegö [16].
The sets X and Y could be allowed to have distinct cardinalities in Theorems 3.1 and 6.1. However, in other results we must assume that these cardinalities coincide.

Determinant formulas
In this section, we derive formulas for determinants of meet matrices. In Theorem 4.1, we present an expression for det(X,Y ) f on arbitrary sets X and Y . Taking .,x n } in Theorem 4.1, we could obtain a formula for the determinant of usual meet matrices (S) f on arbitrary set S (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 Theorems 4.2 and 4.4, respectively, we calculate det(S) f on meet-closed and lower-closed sets S. These formulas are also given in [7]. (4.1) Proof. By Theorem 3.1, Thus by the Cauchy-Binet formula, we obtain Theorem 4.1.
Corollary 4.5 [18,Theorem 2]. Let S be a GCD-closed set of distinct positive integers, and let f be an arithmetical function. Then Corollary 4.6 [1]. Let S be a factor-closed set of distinct positive integers, and let f be an arithmetical function. Then

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 sets X and Y , and in Theorem 5.2 we present an expression for the inverse of (S) f on arbitrary set S. Taking (P, ) = (Z + ,|), we could obtain a formula for the inverse of GCD matrices on arbitrary set S. Formulas for the inverse of meet or GCD matrices on arbitrary set have not previously been presented in the literature. In Theorems 5.3 and 5.5, respectively, we calculate the inverse of (S) f on meet-closed and lower-closed sets S. Similar formulas are given in [12,Theorem 7.1] and [7, Theorem 6].
Proof. It is well known that where α ji is the cofactor of the ji-entry of (X,Y ) f . It is easy to see that α ji = (−1) i+ j det(X j , Y i ) f . By Theorem 4.1, we see that Combining the above equations, we obtain Theorem 5.1.
Proof. Taking X = Y = S in Theorem 5.1, we obtain Theorem 5.2. Theorem

Suppose that S is lower-closed. If (S) f is invertible, then the inverse of
Here μ is the Möbius function of (P, ).
Corollary 5.6 [18,Corollary 1]. Let S be a factor-closed set of distinct positive integers, and let f be an arithmetical function.
Here μ is the number-theoretic Möbius function.

Formulas for join matrices
Let f be a complex-valued function on P. We say that f is a semimultiplicative function if for all x, y ∈ P (see [5]). The notion of a semimultiplicative function arises from the theory of arithmetical functions. Namely, an arithmetical function f is said to be semimultiplicative if f (r) f (s)= f ((r,s)) f ([r,s]) for all r,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 Sections 3, 4, and 5. Throughout this section, f is a semimultiplicative function on P such that f (x) = 0 for all x ∈ P. where We thus obtain (6.2), and applying Theorem 3.1 we obtain (6.3).
From (6.2), we obtain Now, using (6.6) and the formulas of Sections 4 and 5, we obtain formulas for join matrices. We first present formulas for the determinant of join matrices. In Theorem 6.2, we give an expression for det[X,Y ] f on arbitrary sets X and Y . Formulas for the determinant of join or LCM matrices on arbitrary sets have not previously been presented in the literature. In Theorems 6.3 and 6.4, respectively, we calculate det[S] f on meet-closed and lower-closed sets S. Similar formulas are given in [5,Section 5.3]. (6.7) We next derive formulas for inverses of join matrices. In Theorem 6.7, we give an expression for the inverse of [X,Y ] f on arbitrary sets X and Y , and in Theorem 6.8 we give an expression for the inverse of [S] f on arbitrary set S. Taking (P, ) = (Z + ,|) we could obtain a formula for the inverse of LCM matrices on arbitrary set S. Formulas for the inverse of join or LCM matrices on arbitrary set have not previously been presented in the literature. In Theorems 6.9 and 6.10, respectively, we calculate the inverse of [S] f on meet-closed and lower-closed sets S. Similar formulas are given in [5,Section 5.3].
Here μ is the Möbius function of (P, ).
Corollary 6.11 [20,Theorem 2]. Let S be a factor-closed set of distinct positive integers, and let f be a quasimultiplicative arithmetical function such that f (r) = 0 for all r ∈ Z + . If Here μ is the number-theoretic Möbius function.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable: Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009