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We consider any cancellative monoid $M$ equipped with a discrete degree map $deg:M\to R_{\ge0}$ and associated generating function $P(t)=\sum_{m\in M}t^{deg(m)}$, called the growth function of $M$. We also introduce, using some towers of minimal common multiple sets in $M$, another signed generating function $N(t)$, called the skew-growth function of $M$. We show that these functions satisfy the inversion formula $P(t)N(t)=1$. In case the monoid is the set of positive integers with ordinary product structure and the degree map is logarithm function, using the coordinate change $t=exp(-s)$, the inversion formula turns out to be the Euler product formula for the Riemann's zeta function., Comment: 18 pages

We extend a theorem of D. Rees on the existence of Rees valuations of an ideal A of a Noetherian ring to Noetherian multiplicative lattices L. This result also extends a result of D.P. Brithinee. We then apply this to projective equivalence and asymptotic primes of rational powers of A. In particular, it is shown that if L is a Noetherian multiplicative lattice, A∈L, {P1,…,Pr} is the set of centers of the Rees valuations v1,…,vr of A and e is the least common multiple of the Rees numbers e1(A),…,er(A) of A, then Ass(L/An/e)⊆{P1,…,Pr}, where Aβ=⋁{x∈L|v¯A(x)⩾β}. Further, if A⩽̸q for each minimal prime q∈L, then Ass(L/An/e)⊆Ass(L/An/e+k/e) for each n∈N, where k/e is in a certain additive subsemigroup of Q+ which is naturally associated to the set of members of L which are projectively equivalent to A. These latter results are new even in the case of rings and extend results of L.J. Ratliff who gave them for rings in the case that the n/e and k/e are integers.

The denominator of the Hilbert series of a finitely generated R-module M does not always divide the denominator of the Hilbert series of R. For this reason, we define the universal denominator. The universal denominator of a module M is the least common multiple of the denominators of the Hilbert series of all submodules of M. The universal denominator behaves nicely with respect to short exact sequences and tensor products. It also has interesting geometric interpretations. Formulas are given for the universal denominator for rings of invariants. Dixmier gave a conjectural formula for the denominator of the Hilbert series of invariants of binary forms. We show that the universal denominator is actually equal to Dixmier's formula in that case., 19 pages

Let S = { x 1 , … , x n } be a set of n distinct positive integers. Let f be an arithmetical function. Let [ f ( x i , x j ) ] 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 and ( f [ x i , x j ] ) 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. The set S is said to be gcd-closed if ( x i , x j ) ∈ S for all 1 ⩽ i , j ⩽ n . For an integer x, let ν ( x ) denote the number of distinct prime factors of x. In this paper, by using the concept of greatest-type divisor introduced by S. Hong in [Adv. Math. (China) 25 (1996) 566–568; J. Algebra 218 (1999) 216–228], we obtain a new reduced formula for det f [ ( x i , x j ) ] if S is gcd-closed. Then we show that if S = { x 1 , … , x n } is a gcd-closed set satisfying max x ∈ S { ν ( x ) } ⩽ 2 , and if f is a strictly increasing (respectively decreasing) completely multiplicative function, or if f is a strictly decreasing (respectively increasing) completely multiplicative function satisfying 0 f ( p ) ⩽ 1 p (respectively f ( p ) ⩾ p ) for any prime p, then the matrix [ f ( x i , x j ) ] (respectively ( f [ x i , x j ] ) ) defined on S is nonsingular. As a corollary, we show the following interesting result: The LCM matrix ( [ x i , x j ] ) defined on a gcd-closed set is nonsingular if max x ∈ S { ν ( x ) } ⩽ 2 .

Let S = {x1,…,xn} be a set of n distinct positive integers. The matrix [S]n having the least common multiple [xi, xj] of xi and xj as its i, j-entry is called the least common multiple (LCM) matrix on S. A set S is gcd-closed if (xi, xj) ∈ S for 1 ≤ i, j ≤ n. Bourque and Ligh conjectured that the LCM matrix [S]n, defined on a gcd-closed set S, is nonsingular. In this paper we prove that the conjecture is true for n ≤ 7 and is not true for n ≥ 8. So the conjecture is solved completely.

Thin Gaussian groups are a natural generalization of spherical Artin groups, namely groups of fractions of monoids in which the existence of least common multiples is kept as an hypothesis, but the relations between the generators are not supposed to necessarily be of Coxeter type. Here we completely describe the center of thin Gaussian groups by constructing a minimal generating set for the quasi-center. We deduce that every thin Gaussian group is an iterated crossed product of thin Gaussian groups with a cyclic center.

Let e be a real number and S = { x 1 , … , x n } be a set of n distinct positive integers. The set S is said to be gcd-closed (respectively lcm-closed) if ( x i , x j ) ∈ S (respectively [ x i , x j ] ∈ S ) for all 1 ⩽ i , j ⩽ n . The matrix having eth power [ x i , x j ] e of the least common multiple of x i and x j as its i , j -entry is called the eth power least common multiple (LCM) matrix, denoted by ( [ x i , x j ] e ) (or abbreviated by ( [ S ] e ) ). In this paper, we show that for any real number e ⩾ 1 and n ⩽ 7 , the power LCM matrix ( [ x i , x j ] e ) defined on any gcd-closed (respectively lcm-closed) set S = { x 1 , … , x n } is nonsingular. This confirms partially two conjectures raised by Hong in [S. Hong, Nonsingularity of matrices associated with classes of arithmetical functions, J. Algebra 281 (2004) 1–14]. Similar results are established for reciprocal real number power GCD matrices.