A convolution is a mapping 􏰂 of the set 􏰈 + of positive integers into the set 􏰆 (􏰈 + ) of all subsets of 􏰈+ such that, for any n ∈ 􏰈+ , each member of 􏰂(n) is a divisor of n. If 􏰃(n) is the set of all divisors of n, for any n, then 􏰃 is called the Dirichlet’s convolution. Corresponding to any general convolution 􏰂, we can define a binary relation ≤􏰂 on 􏰈+ by “m ≤􏰂 n if and only if m ∈ 􏰂(n)”. It is well known that 􏰈+ has the structure of a distributive lattice with respect to the division order. The division ordering is precisely the partial ordering ≤􏰃 induced by the Dirichlet’s convolution 􏰃. In this paper, we present a characterization for the prime ideals in (􏰈+,≤􏰃) , where 􏰃 is the Dirichlet’s convolution.

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Characterization of Prime Ideals in (Z+,<=D)

A convolution is a mapping $\mathcal{C}$ of the set $Z^{+}$ of positive integers into the set ${\mathscr{P}}(Z^{+})$ of all subsets of $Z^{+}$ such that, for any $n\in Z^{+}$, each member of $\mathcal{C}(n)$ is a divisor of $n$. If $\mathcal{D}(n)$ is the set of all divisors of $n$, for any $n$, then $\mathcal{D}$ is called the Dirichlet's convolution [2]. If $\mathcal{U}(n)$ is the set of all Unitary(square free) divisors of $n$, for any $n$, then $\mathcal{U}$ is called unitary(square free) convolution. Corresponding to any general convolution $\mathcal{C}$, we can define a binary relation $\leq_{\mathcal{C}}$ on $Z^{+}$ by `$m\leq_{\mathcal{C}}n$ if and only if $ m\in \mathcal{C}(n)$'. In this paper, we present a characterization of regular convolution.

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Characterization of regular convolutions

A convolution is a mapping $\mathcal{C}$ of the set $\mathcal{Z^{+}}$ of positive integers into the set $\mathcal{P}(\mathcal{Z^{+}})$ of all subsets of $\mathcal{Z^{+}}$ such that for any $n\in \mathcal{Z^{+}}$, each member of $\mathcal{C}(n)$ is a divisor of $n$. If $D(n)$ is the set of all divisors of $n$, for any $n$, then $D$ is called the Dirichlet's convolution. If $U(n)$ is the set of all unitary(square free) divisors of $n$, for any $n$, then $U$ is called unitary(square free) convolution. Corresponding to any general convolution $\mathcal{C}$, we can define a binary relation $\leq_{\mathcal{C}}$ on $\mathcal{Z^{+}}$ by ` $m\leq_{\mathcal{C}}n $ if and only if $ m\in \mathcal{C}(n)$ '. In this paper, we characterize convolutions $\mathcal{C}$ for which $\leq_{\mathcal{C}}$ is a partial order on $\mathcal{Z^{+}}$ and discuss the various properties of the partial ordered set $(\mathcal{Z^{+}},\leq_{\mathcal{C}})$ in terms of the convolution $\mathcal{C}$.

Partial orders induced by convolutions

A Convolution C is a mapping of the set Z+ of positive integers into the power set P(Z+) such that every member of C(n) is a divisor of n. If for any n, D(n) is the set of all positive divisors of n , then D is called the Dirichlet’s convolution. It is well known that Z+ has the structure of a distributive lattice with respect to the division order. Corresponding to any general convolution C, one can define a binary relation ≤C on Z+ by ‘ m ≤C n if and only if m ∈ C(n)’ . In this paper we characterize Convolutions C which induce partial orders with respect to which Z+ has the structure of a semi lattice or lattice and various lattice theoretic properties are discussed in terms of convolution.

U.M. Swamy

Papers

Partial orders induced by convolutions

Lattice Structures on Z+ Induced by Convolutions