Embedding the unitary divisor meet semilattice in a lattice

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
Full paper (PDF, 245 Kb)

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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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