Geodesic
Structural Properties of a New Class of CM-Lattices
Canadian journal of mathematics, Tome 38 (1986) no. 3, pp. 552-562

Voir la notice de l'article provenant de la source Cambridge University Press

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]. 
Johnson, Johnny A.; Sherette, Gerald R. Structural Properties of a New Class of CM-Lattices. Canadian journal of mathematics, Tome 38 (1986) no. 3, pp. 552-562. doi: 10.4153/CJM-1986-027-0
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