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

• Let (P, ≤P) and let (Q, ≤Q) be

disjoint quasiordered sets.

• The cardinal sum is defined to be

the quasiordered set (P ∪ Q, ≤P ∪ ≤Q). Q P 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 is called a merging of P and Q if ≤R,S:= {≤P, ≤Q, R, S} is a quasiorder again.

• A merging (R, S) of P and Q is

said to be proper if R ∩ S−1 is empty. Q P P Q ≤P R ≤Q S

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 pR is an order filter in Q, and for every q ∈ Q the column qR is an order ideal in P; (b) R is an order ideal in the quasiordered set P × Qd; (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 × Qd, (2) S−1 is an order filter in P × Qd, (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

a b 1 2 a, 1 b, 2 a 1 b 2

a b 1 2 a × × b × 1 × × 2 × a b 1 2 a × × × × b × × 1 × × × × 2 × × a b 1 2 a × × × b × × 1 × × × 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 (R1, S1) ≤ (R2, S2) :⇐ ⇒ R1 ⊆ R2 and S1 ⊇ S2. The indicated expressions for infimum and supremum are given by:

• t∈T

(Rt, St) =

t∈T

Rt,

• t∈T

St

• ,
• t∈T

(Rt, St) =

t∈T

Rt,

• t∈T

St

• .

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 P × Q

Ψ ⊒ ⊒ =: M The representing context M of the lattice M of all mergings. Thereby, ⊑ denotes the order on P × Qd. Hence, we have that (p1, q1) ⊑ (p2, q2) iff p1 ≤ p2 and q1 ≥ q2.

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 (p1, q1) Ψ (p2, q2) :⇐ ⇒

• q1 ≤ q2 ⇒ p1 ≤ p2, and

p1 ≥ p2 ⇒ q1 ≥ q2 An isomorphism ϕ : M → B(M) is given by (R, S) − →

• R ⊎ (S−1)∁, S−1 ⊎ R∁

.

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

Contextual Representation

P × Q P × Q P × Q P × Q

Ψ• ⊒ ⊒ =: M• The representing context M• of the lattice M• of all proper

• mergings. Thereby, ⊑ denotes the order on P × Qd 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 (p1, q1) Ψ• (p2, q2) :⇐ ⇒

• q1 ≤ q2 ⇒ p1 < p2, and

p1 ≥ p2 ⇒ q1 > q2 An isomorphism ϕ : M• → B(M•) is given by (R, S) − →

• R ⊎ (S−1)∁, S−1 ⊎ R∁

.

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 P × Q

⊒ ⊒ ⊒ =: M•

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

Generalisations

• Let P := (Pt, ≤t)t∈T be a family of pairwise disjoint quasi-
• rdered sets.
• Let ≤ be a fixed linear order on T.
• We put P :=

t∈T Pt. We call R ⊆ P × P a merging of P if

it is a quasiorder on P that satisfies Rt = ≤t for every t ∈ T.

• Thereby, for s, t ∈ T and X ⊆ P × P we put

Xs,t := X ∩ Ps × Pt and Xt := Xt,t.

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

Generalisations

• Let P := (Pt, ≤t)t∈T be a family of pairwise disjoint quasi-
• rdered sets.
• Let ≤ be a fixed linear order on T.
• We put P :=

t∈T Pt. We call R ⊆ P × P a merging of P if

it is a quasiorder on P that satisfies Rt = ≤t for every t ∈ T.

• Thereby, for s, t ∈ T and X ⊆ P × P we put

Xs,t := X ∩ Ps × Pt and Xt := Xt,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 Rs,t ∩ R−1

t,s is empty.

• For two mergings X and Y of P we define

X ≤ Y :⇐ ⇒    Xs,t ⊆ Ys,t for s < t, Xs,t ⊇ Ys,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?