The dual notion of an ideal, i. Some authors use the term ideal to mean a lower set, i. Frink ideals , pseudoideals and Doyle pseudoideals are different generalizations of the notion of a lattice ideal. An ideal or filter is said to be proper if it is not equal to the whole set P. The smallest ideal that contains a given element p is a principal ideal and p is said to be a principal element of the ideal in this situation. An important special case of an ideal is constituted by those ideals whose set-theoretic complements are filters, i.
Such ideals are called prime ideals. Also note that, since we require ideals and filters to be non-empty, every prime ideal is necessarily proper. For lattices, prime ideals can be characterized as follows:. For a complete lattice the further notion of a completely prime ideal is meaningful. It is defined to be a proper ideal I with the additional property that, whenever the meet infimum of some arbitrary set A is in I , some element of A is also in I. So this is just a specific prime ideal that extends the above conditions to infinite meets. The existence of prime ideals is in general not obvious, and often a satisfactory amount of prime ideals cannot be derived within ZF Zermelo—Fraenkel set theory without the axiom of choice.
This issue is discussed in various prime ideal theorems , which are necessary for many applications that require prime ideals. An ideal I is maximal if it is proper and there is no proper ideal J that is a strictly greater set than I. Likewise, a filter F is maximal if it is proper and there is no proper filter that is strictly greater. When a poset is a distributive lattice , maximal ideals and filters are necessarily prime, while the converse of this statement is false in general.
In Boolean algebras, the terms prime ideal and maximal ideal coincide, as do the terms prime filter and maximal filter.
There is another interesting notion of maximality of ideals: Consider an ideal I and a filter F such that I is disjoint from F. We are interested in an ideal M that is maximal among all ideals that contain I and are disjoint from F. In the case of distributive lattices such an M is always a prime ideal. A proof of this statement follows. However, in general it is not clear whether there exists any ideal M that is maximal in this sense. Yet, if we assume the Axiom of Choice in our set theory, then the existence of M for every disjoint filter—ideal-pair can be shown. Format: Paperback. A set with, at minimum, one binary operation is a groupoid.
If a situation involves an equivalence relation or some sort of symmetry, some sort of groupoid applies. If the set has, at minimum, two binary operations, and one operation distributes over the other, you have a ringoid.
Ringoids, which include the real field we all use every day, tell us much about number systems. Let there be a groupoid. Denote its single binary operation by concatenation. Let that operation commute and associate. So far, we have a commutative semigroup. With that seemingly trivial axiom we turn a corner, farewell the groupoids, and find ourselves among the semilattices. What you now have is a lattice, of which the best known example is Boolean algebra which requires added axioms.
- Introduction to Lattices and Order : B. A. Davey : !
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More generally, most logics can be seen as interpretations of bounded lattices. Given any relation of partial or total order, the corresponding algebra is lattice theory. Nevertheless, far fewer mathematicians specialize in lattices than in groupoids and ringoids. Davey and Priestley has become the classic introduction to lattice theory in our time.
Introduction To Lattices And Order 2nd Edition
Sad to say, it has little competition. It is a bit harder than I would prefer, and the authors do not say enough about the value of lattice theory for nonclassical logic. Their book is a classic nonetheless, and here's hoping that Gian Carlo Rota was right when he said that the 21st century shall be the century of lattices triumphant. Lattice theory is largely due to the work of the American Garrett Birkhoff, writing in the s. He gets my vote for the greatest American mathematician of all time.
This book presents an excellent introduction to the subject, but also goes beyond that, presenting with a fair amount of the detail the theory of Priestley representation. The excercises start at the basic level of checking the understanding of definitions, allowing the reader to build confidence out of the practice. The fact that Priestley herself co-authored it is definitely a plus.
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Embedding Ordered Sets into Distributive Lattices
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Introduction to Lattices and Order, Second Edition
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