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Structural Properties of a New Class of CM-Lattices

Published online by Cambridge University Press:  20 November 2018

Johnny A. Johnson  and
Gerald R. Sherette
Johnny A. Johnson
Affiliation:
University of Houston-University Park, Houston, Texas
Gerald R. Sherette
Affiliation:
University of Houston-University Park, Houston, Texas
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1. Introduction. In this paper we introduce and study a class of multiplicative lattices called q-lattices. A q-lattice is a principally generated multiplicative lattice in which each principal element is compact. One of our main objectives is to characterize principal elements in these lattices (We note that Noether lattices and r-lattices are q-lattices [1, Theorem 2.1] and so our results apply to these two types of lattices). Among other things we determine necessary and sufficient conditions for globalizing local results in q-lattices. We then apply localization to establish some properties of principal elements in general q-lattices. Conditions equivalent to an element being principal are known for several different classes of multiplicative lattices. For example, Bogart [2] showed that if the lattice is modular, weak principal is equivalent to principal; Johnson and Lediaev pointed out that for Noether lattices, meet principal is equivalent to principal [5]; and, in an r-lattice, an element is principal if and only if it is compact and weak meet principal [6].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 3 , 01 June 1986 , pp. 552 - 562
Copyright
Copyright © Canadian Mathematical Society 1986

References

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