Lattices

R. Kochendörffer
Additional contact information
R. Kochendörffer: University of Dortmund

Chapter 11 in Introduction to Algebra, 1972, pp 379-406 from Springer

Abstract: Abstract If A and B are subsets of a given set M then so are A ⋂ B and A ⋃ B. So the set of all subsets of M can be regarded as an algebraic system with the operations ⋂ and ⋃ If A and B are subgroups of a given group G, then A ⋂ B is again a subgroup of G. Apart from trivial cases, however, the set theoretical union of A and B is not a subgroup of G. But if A ⋃B is defined as the subgroup generated by A and B, then the set of all subgroups of G can again be considered as an algebraic system with the operations ⋂ and ⋃. The same applies to the set of all normal subgroups of G. Similarly, the subfields of a given field form an algebraic system with the operations ⋂ and ⋃ where A ⋃B means the subfield generated by the subfields A and B. Other examples of such algebraic systems arise when we consider the set of all subspaces of a given vector space or the set of all ideals of a given ring. This suggests the study of algebraic systems with two operations ⋂ and ⋃ which have some but in general not all the properties of the set theoretical intersection and union.

Date: 1972
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-009-8179-9_11

Ordering information: This item can be ordered from
http://www.springer.com/9789400981799

DOI: 10.1007/978-94-009-8179-9_11

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().