Merging Ordered Sets Christian Meschke Institut f ur Algebra - PowerPoint PPT Presentation

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Merging Ordered Sets Christian Meschke Institut f¨ ur Algebra Technische Universit¨ at Dresden February 23, 2011

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Survey Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Setting Q P • Let ( P , ≤ P ) and let ( Q , ≤ Q ) be disjoint quasiordered sets. ≤ P ∅ P • The cardinal sum is defined to be the quasiordered set ≤ Q Q ∅ ( P ∪ Q , ≤ P ∪ ≤ Q ) .

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Mergings and Proper Mergings • A pair ( R , S ) with R ⊆ P × Q and S × Q × P P Q is called a merging of P and Q if ≤ P P R ≤ R , S := {≤ P , ≤ Q , R , S } is a quasiorder again. ≤ Q Q S • A merging ( R , S ) of P and Q is said to be proper if R ∩ S − 1 is empty.

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Bonds between contraordinal scales Proposition Let ( P , ≤ P ) and ( Q , ≤ Q ) be quasiordered sets, and let R ⊆ P × Q . Then the following three statements are equivalent: (a) For every p ∈ P the row p R is an order filter in Q , and for every q ∈ Q the column q R is an order ideal in P ; (b) R is an order ideal in the quasiordered set P × Q d ; (c) R is a bond from ( P , P , � P ) to ( Q , Q , � Q ).

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Characterisation Proposition Let ( P , ≤ P ) and ( Q , ≤ Q ) be disjoint quasiordered sets, and let R ⊆ P × Q and let S ⊆ Q × P . Then the pair ( R , S ) is a merging if and only if all of the following four properties are satisfied: (1) R is an order ideal in P × Q d , (2) S − 1 is an order filter in P × Q d , (3) R ◦ S ⊆ ≤ P , (4) S ◦ R ⊆ ≤ Q . Furthermore, ≤ R , S is antisymmetric iff both, ≤ P and ≤ Q are antisymmetric and the intersection R ∩ S − 1 is empty.

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations The situation for posets When P and Q are posets, the notion of proper mergings seems to be more natural: Corollary Let P and Q be disjoint partially ordered sets, let R ⊆ P × Q and let S ⊆ Q × P . Then ( P ∪ Q , ≤ R , S ) is a partially ordered set again if and only if ( R , S ) is a proper merging.

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations An example b , 2 2 b 2 b a 1 a , 1 a 1 a b 1 2 a b 1 2 a b 1 2 a a a × × × × × × × × × b b b × × × × × 1 1 1 × × × × × × × × × 2 2 2 × × × ×

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Lattices of mergings, part 1 Let P and Q be disjoint quasiordered sets. (i) Then the set M of all mergings of P and Q forms a complete lattice if one orders it by ( R 1 , S 1 ) ≤ ( R 2 , S 2 ) : ⇐ ⇒ R 1 ⊆ R 2 and S 1 ⊇ S 2 . The indicated expressions for infimum and supremum are given by: � � � � � ( R t , S t ) = R t , S t , t ∈ T t ∈ T t ∈ T � � � � � ( R t , S t ) = R t , S t . t ∈ T t ∈ T t ∈ T

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Lattices of Mergings, part 2 (ii) The set M • of all proper mergings forms a complete sublattice of M . (iii) The least (proper) merging is ( ∅ , Q × P ), whereas the greatest one is ( P × Q , ∅ ).

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Corollary Let P and Q be disjoint quasiordered sets, let X ⊆ P × Q and let Y ⊆ Q × P . Then the set of all (proper) mergings ( R , S ) with X ⊆ R and Y ⊆ S is empty, or forms an interval in M (in M • ).

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Questions • What are contextual representations of M and M • . • What are contextual represenations of the lattices of (proper) extensions. • How can the “non-disjoint” case be described? • What are possible applications? • How can one generalise the situation to the case of more than two quasiordered sets?

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Questions • What are contextual representations of M and M • . • What are contextual represenations of the lattices of (proper) extensions. • How can the “non-disjoint” case be described? • What are possible applications? • How can one generalise the situation to the case of more than two quasiordered sets?

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Contextual Representation P × Q P × Q �⊒ P × Q Ψ =: M �⊒ P × Q The representing context M of the lattice M of all mergings. Thereby, ⊑ denotes the order on P × Q d . Hence, we have that ( p 1 , q 1 ) ⊑ ( p 2 , q 2 ) iff p 1 ≤ p 2 and q 1 ≥ q 2 .

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Contextual Representation Theorem (i) The lattice M of all mergings of P and Q is isomorphic to the concept lattice of the context M displayed above. Thereby, the relation Ψ from the upper left quadrant is given by � q 1 ≤ q 2 ⇒ p 1 ≤ p 2 , and ( p 1 , q 1 ) Ψ ( p 2 , q 2 ) : ⇐ ⇒ p 1 ≥ p 2 ⇒ q 1 ≥ q 2 An isomorphism ϕ : M → B ( M ) is given by � R ⊎ ( S − 1 ) ∁ , S − 1 ⊎ R ∁ � ( R , S ) �− → .

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Contextual Representation P × Q P × Q �⊒ Ψ • P × Q =: M • �⊒ P × Q The representing context M • of the lattice M • of all proper mergings. Thereby, ⊑ denotes the order on P × Q d again.

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Contextual Representation Theorem (ii) The lattice M • of all proper mergings is isomorphic to the concept lattice of the context M • displayed above. Thereby, the relation Ψ • from the upper left quadrant is given by � q 1 ≤ q 2 ⇒ p 1 < p 2 , and ( p 1 , q 1 ) Ψ • ( p 2 , q 2 ) : ⇐ ⇒ p 1 ≥ p 2 ⇒ q 1 > q 2 An isomorphism ϕ : M • → B ( M • ) is given by � R ⊎ ( S − 1 ) ∁ , S − 1 ⊎ R ∁ � ( R , S ) �− → .

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations some remarks • One can easily show that Ψ and Ψ • are self-bonds of the contraordinal scale ( P × Q , P × Q , �⊒ ). • Furthermore, it follows that Ψ • ⊆ Ψ. • If both, P and Q are chains, it follows that �⊒ = Ψ • . Then P × Q P × Q �⊒ �⊒ P × Q =: M • �⊒ P × Q

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Generalisations • Let P := ( P t , ≤ t ) t ∈ T be a family of pairwise disjoint quasi- ordered sets. • Let ≤ be a fixed linear order on T . • We put P := � t ∈ T P t . We call R ⊆ P × P a merging of P if it is a quasiorder on P that satisfies R t = ≤ t for every t ∈ T . • Thereby, for s , t ∈ T and X ⊆ P × P we put X s , t := X ∩ P s × P t and X t := X t , t .

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Generalisations • Let P := ( P t , ≤ t ) t ∈ T be a family of pairwise disjoint quasi- ordered sets. • Let ≤ be a fixed linear order on T . • We put P := � t ∈ T P t . We call R ⊆ P × P a merging of P if it is a quasiorder on P that satisfies R t = ≤ t for every t ∈ T . • Thereby, for s , t ∈ T and X ⊆ P × P we put X s , t := X ∩ P s × P t and X t := X t , t .

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Generalisations • A merging R of P is called proper if for all s < t from T the intersection R s , t ∩ R − 1 t , s is empty. • For two mergings X and Y of P we define  X s , t ⊆ Y s , t for s < t ,  X ≤ Y : ⇐ ⇒ X s , t ⊇ Y s , t for s > t . 

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations

Mergings and Proper Mergings Lattices of (Proper) Mergings Contextual Representation Generalisations Questions? Thank you for your attention. Questions?