Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T00:30:33.777Z Has data issue: false hasContentIssue false
  • English
  • Français

The tensor product of distributive lattices: structural results

Published online by Cambridge University Press:  20 January 2009

Andrew M. Bell ,
Michael R. Brown  and
Grant A. Fraser
Andrew M. Bell
Affiliation:
University of Illinois, Urbana, Illinois 61801
Michael R. Brown
Affiliation:
California State UniversityHayward, California 94542
Grant A. Fraser
Affiliation:
California State UniversityLos Angeles, California 90032
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The tensor product A ⊗ B of the distributive lattices A and B was first investigated by Fraser in [4] and [5]. In this paper, we present some results relevant to the structure and construction of this tensor product. In particular, we establish a sufficient condition for join-irreducibility in the tensor product and show that this condition characterizes join-irreducibility in the case that A and B satisfy the descending chain condition. We also show that if A and B satisfy the descending chain condition then so does A ⊗ B; this insures the compact generation of A ⊗ B by its join-irreducibles. We conclude with some examples and applications of our results to the tensor product of finite distributive lattices.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 27 , Issue 3 , October 1984 , pp. 237 - 245
Copyright
Copyright © Edinburgh Mathematical Society 1984

References

REFERENCES

1Balbes, R., Projective and injective distributive lattices, Pacific J. Math. 21 (1967), 405–420.CrossRefGoogle Scholar
2Dilworth, R. P., A decomposition theorem for partially ordered sets, Annals of Math. 51 (1950), 161–166CrossRefGoogle Scholar
3Dilworth, R. P. and Crawley, P., Algebraic Theory of Lattices (Prentice-Hall, Englewood Cliffs, N.J., 1973).Google Scholar
4Fraser, G. A., The tensor product of distributive lattices, Proc. Edinburgh Math. Soc. 20 (1976), 121–131.CrossRefGoogle Scholar
5Fraser, G. A., The tensor product of distributive lattices II, Proc. Edinburgh Math. Soc. 20 (1977), 355–360.CrossRefGoogle Scholar
6Grêtzer, G., General Lattice Theory (Academic Press, New York, 1978).CrossRefGoogle Scholar
7Kucera, T. G. and Sands, B., Lattices of lattice homomorphisms, Algebra Universalis 8 (1978), 180–190.CrossRefGoogle Scholar