**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

**Download full paper: PDF, 245 Kb**

## 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

### References

- Aigner, M., (1979) Combinatorial Theory, Springer.
- Altinisik, E., Sagan, B. E. & Tuglu, N. (2005) GCD matrices, posets, and nonintersecting paths, Linear Multilinear Algebra 53, 75–84.
- Birkhoff, G. (1979) Lattice theory, Corrected reprint of the 1967 third edition, American Mathematical Society Colloquium Publications, 25. American Mathematical Society, 1979.
- Cohen, E., (1960) Arithmetical functions associated with the unitary divisors of an integer, Math. Z., 74, 66–80.
- Crawley, P. & Dilworth, R. P. (1973) Algebraic Theory of Lattices, Prentice-Hall.
- Dugundji, J. (1966) Topology, Allyn and Bacon.
- Hansen, R. T. & Swanson, L. G. (1979) Unitary divisors, Math. Mag. 52, 217–222.
- Haukkanen, P. (2012) Extensions of the class of multiplicative functions. East-West J. Math. 14 (2), 101–113.
- Haukkanen, P., Ilmonen, P., A. Nalli, Ayse & J. Sillanp¨a¨a (2010) On unitary analogs of GCD reciprocal LCM matrices. Linear Multilinear Algebra 58(5–6), 599–616.
- Hong, S. & Sun, Q. (2004) Determinants of matrices associated with incidence functions on posets. Czechoslovak Math. J. 54 (129), no. 2, 431–443.
- Ilmonen, P., Haukkanen, P. & Merikoski, J. (2008) On eigenvalues of meet and join matrices associated with incidence functions, Linear Algebra Appl. 429, 859–874.
- Korkee, I. (2006) On meet and join matrices associated with incidence functions, Ph.D. Thesis, Acta Universitatis Tamperensis 1149, Tampere Univ. Press.
- Mattila, M. (2015) On the eigenvalues of combined meet and join matrices, Linear Algebra Appl. 466, 1–20.
- McCarthy, P. J. (1986) Introduction to Arithmetical Functions, Springer.
- Rearick, D. (1966) Semi-multiplicative functions. Duke Math. J. 33, 4–53.
- Sandor J. & Crstici, B. (2004) Handbook of Number Theory II, Kluwer Academic.
- Selberg, A. (1977) Remarks on multiplicative functions. In: Number Theory Day. Proc. Conf., Rockefeller Univ., New York, 1976, pp. 232–241. Springer, Berlin.
- Sivaramakrishnan, R. (1989) Classical Theory of Arithmetic Functions, Marcel Dekker, Inc.
- Stanley, R. P. (1986) Enumerative Combinatorics, Vol. 1, Wadsworth and Brooks/Cole.
- Szasz, G. (1963) Introduction to Lattice Theory, Third Ed., Academic Press, Budapest.
- Toth, L. (1989) The unitary analogue of Pillai’s arithmetical function. Collect. Math. 40(1), 19–30.
- Vaidyanathaswamy, R. (1931) The theory of multiplicative arithmetic functions, Trans. Amer. Math. Soc., 33, 579–662.

## Related papers

## Cite this paper

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