**Pentti Haukkanen**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 22, 2016, Number 3, Pages 68—78

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## Details

### Authors and affiliations

Pentti Haukkanen

*School of Information Sciences
FI-33014 University of Tampere, Finland*

### Abstract

A positive divisor *d* of a positive integer *n* is said to be a unitary divisor of *n* if (*d, n/d*) = 1. The set of positive integers is a meet semilattice under the unitary divisibility relation but not a lattice since the least common unitary multiple (lcum) does not always exist. This meet semilattice can be embedded to a lattice; two such constructions have hitherto been presented in the literature. Neither of them is distributive nor locally finite. In this paper we embed this meet semilattice to a locally finite distributive lattice. As applications we consider semimultiplicative type functions, meet and join type matrices and the Möbius function of this lattice.

### Keywords

- Unitary divisor
- Meet semilattice
- Distributive lattice
- Semimultiplicative function
- Meet matrix
- Möbius function

### AMS Classification

- 06A12
- 06D99
- 11A25
- 11A51
- 11C20
- 15B36

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## Cite this paper

APAHaukkanen, P. (2016). Embedding the unitary divisor meet semilattice in a lattice, Notes on Number Theory and Discrete Mathematics, 22(3), 68-78.

ChicagoHaukkanen, Pentti “Embedding the Unitary Divisor Meet Semilattice in a Lattice.” Notes on Number Theory and Discrete Mathematics 22, no. 3 (2016): 68-78.

MLAHaukkanen, Pentti “Embedding the Unitary Divisor Meet Semilattice in a Lattice.” Notes on Number Theory and Discrete Mathematics 22.3 (2016): 68.78. Print.