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 [2]) and therefore the majority of concepts and results have an analogue (or an application) in commutative rings (see [1]).

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 [3]). 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.

References

[1]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
[2]	R.P. Dilworth, "Abstract commutative ideal theory" Pacific J. Math. , 12 (1962) pp. 481–498
[3]	K. Bogart, "Nonimbeddable Noether lattices" Proc. Amer. Math. Soc. , 22 : 1 (1969) pp. 129–133
[4a]	K. Bogart, "Structure theorems for regular local Noether lattices" Michigan J. Math. , 15 : 2 (1968) pp. 167–176
[4b]	K. Bogart, "Distributive Noether lattices" Michigan J. Math. , 16 : 3 (1969) pp. 215–223
[5]	T.S. Fofanova, "General theory of lattices" , Ordered sets and lattices , 3 , Saratov (1975) pp. 22–40 (In Russian)

Comments

For a fixed $ a \in L $, the two functions $ x \mapsto a : x $, $ a \cdot x : L \rightarrow F $ form a covariant Galois connection. If $ a \cdot b = a \wedge b $, then $ L $ is a complete Heyting algebra with $ a : b = a \Rightarrow b $.

The study of multiplicative lattices began with a celebrated paper of M. Ward and R.P. Dilworth [a1], who called them residuated lattices; this name focuses attention on the residuation operation $ a : b $ characterized by

$$ c \leq a : b \ \textrm{ if and only if } \ \ c \cdot b \leq a . $$

The existence of this operation, given the multiplication, is equivalent to the distributive law given in the main article above. Many authors (e.g. [a2]) have sought to dispense with the requirement that the multiplication be commutative (in this case, both left and right distributive laws must be assumed, and there are two different residuation operations) and/or that the top element of the lattice be a unit for multiplication; some have been dropped the associativity of the multiplication (cf. [a3]), but relatively little work has been done on the non-associative case. Recently much interest has been shown in the quantales introduced by C.J. Mulvey [a4], where the multiplication is non-commutative but is generally assumed to be idempotent (i.e. to satisfy the identity $ a \cdot a = a $) and to have the top element as a one-sided unit. In the commutative case, the assumption of idempotency forces the multiplication to coincide with the lattice-theoretic meet (and forces the underlying lattice to be a frame, cf. Locale); thus idempotent quantales may be viewed as a "non-commutative" generalization of topological spaces, cf. [a5], [a6]. These concepts find application in the representation theory of non-commutative $ C ^ {*} $- algebras (see $ C ^ {*} $- algebra; the lattice of closed right ideals of a $ C ^ {*} $- algebra is an example of a quantale).

References

[a1]	M. Ward, R.P. Dilworth, "Residuated lattices" Trans. Amer. Math. Soc. , 45 (1939) pp. 335–354
[a2]	R.P. Dilworth, "Non-commutative residuated lattices" Trans. Amer. Math. Soc. , 46 (1939) pp. 426–444
[a3]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1967)
[a4]	C.J. Mulvey, "&" Suppl. ai Rend. Circ. Mat. Palermo (2) , 12 (1986) pp. 99–104
[a5]	S.B. Niefield, K.I. Rosenthal, "Constructing locales from quantales" Math. Proc. Cambridge Philos. Soc. , 104 (1988) pp. 215–234
[a6]	F. Borceux, G. van den Bossche, "An essay in non-commutative topology" Topology Appl. , 31 (1989) pp. 203–223
[a7]	T.S. Blyth, M.F. Janowitz, "Residuation theory" , Pergamon (1972)
[a8]	K. Keimel, "A unified theory of minimal prime ideals" Acta Math. Acad. Scient. Hung. , 23 (1972) pp. 51–69