Duality (lattice theory)

With duality or dualism is in mathematics a double reciprocal assignment shall apply:

On the one hand , it affects structures , their operations , relations or elements
On the other hand , it concerns formulas with which the structures are described.

The double assignment is made in such a way that there is a clear connection between the validity of the corresponding formulas in the corresponding structures . In association theory, the following applies:

Sentences of association theory become sentences of association theory again through dualization.

The double description of lattices corresponds to a twofold possibility of defining the dualization: If one considers lattices as algebraic structures , then there is a duality between the two operations: and are exchanged. If you consider them as organizational structures , then and are dual to one another. The assignment between the formulas is given in both cases by interchanging the corresponding symbols. These two dualities are compatible due to the mutual definability . One can therefore always use the duality that suits the situation better. ${\ displaystyle \ sqcap}$ ${\ displaystyle \ sqcup}$ ${\ displaystyle \ leq}$ ${\ displaystyle \ geq}$

Clarification

Algebraic definition of duality

The two associations are dual to each other (but obviously not isomorphic).

If you swap the two links and in an association , you get a new structure . One calls the dual structure too . ${\ displaystyle V}$ ${\ displaystyle \ sqcap}$ ${\ displaystyle \ sqcup}$ ${\ displaystyle W}$ ${\ displaystyle W}$ ${\ displaystyle V}$

If you take any formula from the language of association theory and insert the two signs “ ” and “ ” alternately for each other, then the resulting formula is called the dual formula of . ${\ displaystyle \ varphi}$ ${\ displaystyle \ sqcap}$ ${\ displaystyle \ sqcup}$ ${\ displaystyle {\ widehat {\ varphi}}}$ ${\ displaystyle \ varphi}$

Obviously, in the structure that is dual to the association , exactly the dual ones to those in the applicable formulas apply. Since the dual formula occurs in the definition of an association for every formula, it follows that there is also an association which is referred to as the association that is too dual . ${\ displaystyle W}$ ${\ displaystyle V}$ ${\ displaystyle V}$ ${\ displaystyle W}$ ${\ displaystyle V}$

Order definition of duality

In the case of partial orders , the transition to the dual partial order is defined by the fact that the order relation is "reversed", ie. H. it is everywhere replaced by and vice versa. Here one defines dual formulas by the mutual substitution of the two relation signs " " and " ". It is clear that the dual formulas apply in the dual partial order and in particular that it is a partial order at all. ${\ displaystyle \ leq}$ ${\ displaystyle \ geq}$ ${\ displaystyle \ leq}$ ${\ displaystyle \ geq}$

Equivalence of definitions in associations

Both definitions can be used in one association. This does not create any problems: When transitioning between the partial order of the lattice and the lattice operators, the dual formula is used in the second part on each side:

${\ displaystyle v \ leq w \ quad \ iff \ quad v \ sqcap w = v}$
${\ displaystyle v \ geq w \ quad \ iff \ quad v \ sqcup w = v}$

It is therefore irrelevant whether the dualization is carried out with the rule for the lattice operators or with the rule for the order relation: one gets "the same" result in each case.

Inferences

If the dual association is closed , then it is obviously dual closed : by swapping twice you get the formulas from which you started. One says in a simplistic way: and are dual to one another . The same applies to formulas that are dual to one another . ${\ displaystyle W}$ ${\ displaystyle V}$ ${\ displaystyle V}$ ${\ displaystyle W}$ ${\ displaystyle V}$ ${\ displaystyle W}$

From this follows the semantic form of the duality principle :

If one formula applies in all associations, then its dual formula also applies in all associations.

This principle can also be formulated syntactically :

If a formula is to be inferred from the association axioms, then its dual formula must also be inferred.

This is of course due to the fact that the dual formula also appears as an axiom for every axiom.

The modularity law is self - dual and the two distributive laws are dual to each other.

The complementary laws are also dual to one another. However, it applies to the special elements "0" and "1" that they swap their roles. Therefore you have to swap the formulas in which the names of these elements occur if you reverse the order. So the dual formula to is . ${\ displaystyle a \ sqcap {\ overline {a}} = 0}$ ${\ displaystyle a \ sqcup {\ overline {a}} = 1}$

But then applies

If a formula applies in all modular or in all distributive lattices or in all Boolean algebras , then the dual formula also applies in the corresponding lattices.

Extensions of the term

With the help of the duality principle, one can always argue in a proof: “ The dual assertion follows dual .” One speaks of the “ dual proof ” if all the formulas that occur are replaced by the dual formulas.

The term is also extended to structures in an obvious way. As an example: Dual formulas are used
in the definition of “ filter ” and “ ideals ”. That is why filters and ideals are called mutually dual structures . If one has proven any theorem about filters, then the dual theorem about ideals holds and vice versa.

Sometimes the dual structure doesn't even get its own name. So one speaks simply of a semi-modular association , if one had to say more precisely "upward-semi-modular" association. If one wants to speak of a "downward semi-modular" association, one paraphrases: "an association for which the dual association is semimodular".

Dual isomorphisms

Hasse diagram showing all combinations formed from two elementary statements (the free Boolean algebra generated by 2 elements ). The blue arrows mark the transition to the dual element. (The generating elements and their complements are mapped onto themselves.)

The duality induces a 1-1 mapping between two associations, which is of course not an association homomorphism. The terms anti-isomorphism or dual isomorphism and antitone , anti-monotonic or order-reversing mapping are used here.

If an association is isomorphic to its image, then the two are usually identified. In this case one speaks of the element in the association which is dual to a .

Dual isomorphisms are generally not clearly defined.

Examples

For with the natural order every strictly monotonically decreasing surjective function can be interpreted as a dual isomorphism. ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}$

If one assumes in propositional logic that the atomic statements remain unchanged due to the duality, then the assignment of the dual statement is unambiguous in the free Boolean algebra that is generated. That is why one also speaks of " the dual statement".

swell

Rudolf Berghammer: Orders, associations and relations with applications . 2nd Edition. Springer + Vieweg, Wiesbaden 2012, ISBN 978-3-658-00618-1 .
Garrett Birkhoff : Lattice Theory . 3. Edition. AMS, Providence, RI 1973, ISBN 0-8218-1025-1 .
Helmuth Gericke: Theory of Associations . Bibliographical Institute, Mannheim 1963.
Hans Hermes : Introduction to Association Theory . 2nd Edition. Springer-Verlag, Berlin - Heidelberg 1967.

Individual evidence

↑ Both terms in H. Gericke, Theory of Associations, p. 153

[1] Both terms in H. Gericke, Theory of Associations, p. 153