# Multiplicative lattice

A complete lattice $L = \langle L , \lor , \wedge \rangle$ with an additional commutative and associative binary operation, called multiplication (and denoted by $\cdot$) such that the largest element of the lattice acts as the multiplicative identity and such that

$$a \cdot \left ( \lor _ {\alpha \in J } b _ \alpha \right ) = \ \lor _ {\alpha \in J } a \cdot b _ \alpha$$

for any $a , b _ \alpha \in L$ and an arbitrary index set $J$. The theory of multiplicative lattices arose as a result of the application of lattice-theoretic methods in the study of lattices of ideals of commutative rings (see ) and therefore the majority of concepts and results have an analogue (or an application) in commutative rings (see ).

Let $L$ be a multiplicative lattice and let $a , b \in L$; then one defines $a : b = \lor \{ {x } : {x \in L, x \cdot b \leq a } \}$. An element $e \in L$ is called $\lor$- principal (respectively, $\wedge$- principal) if $( a \lor ( b \cdot e ) ) : e = ( a : e ) \lor b$( respectively, $( a \wedge ( b : e ) ) \cdot e = ( a \cdot e ) \wedge b$) for any $a , b \in L$; an element which is simultaneously $\lor$- principal and $\wedge$- principal is called principal. A Noether lattice is a modular multiplicative lattice satisfying the ascending chain condition and in which each element is a union of principal elements (cf. also Modular lattice). A complete lattice is called a module over a multiplicative lattice $L$ if for any $\lambda \in L$, $a \in M$ a product $\lambda a \in M$ is defined, where

$$( \lambda _ {1} \lambda _ {2} ) a = \ \lambda _ {1} ( \lambda _ {2} a ) ,\ \ 1 a = a ,\ 0 _ {L} a = 0 _ {M} ,$$

$$\left ( \lor _ {\alpha \in I } \lambda _ \alpha \right ) \left ( \lor _ {\beta \in J } a _ \beta \right ) = \ \lor _ {\alpha , \beta } \lambda _ \alpha a _ \beta$$

(here $\lambda _ {1} , \lambda _ {2} , \lambda _ \alpha \in L$, $a , a _ \beta \in M$, $1$ is the largest element in $L$ and $0 _ {L} , 0 _ {M}$ are the zeros in $L$ and $M$, respectively).

The best studied class of multiplicative lattices are the Noether lattices. Here it is possible to distinguish the following directions. 1) Questions of the representation of a Noether lattice as the lattice of ideals of a suitable Noetherian ring (it is known that the lattice of ideals of a Noetherian ring is a Noether lattice; however, there are Noether lattices that cannot even be imbedded in the lattice of ideals of a Noetherian ring ). 2) The study of Noetherian modules over a multiplicative lattice. 3) The study of concepts and properties from the theory of ideals of Noetherian rings that translate to Noether lattices (the notions of prime and primary elements, of dimension, of a proper maximal element, and of semi-local and local lattices). The distributive regular Noether multiplicative lattices have been described

(cf. also Distributive lattice; Regular lattice). A theory of localization and associated prime elements has been constructed for a class of multiplicative lattices considerably broader than Noether lattices, including lattices of ideals of arbitrary commutative rings.

How to Cite This Entry:
Multiplicative lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplicative_lattice&oldid=47935
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article