The lattice of quasiorder lattices of algebras on a finite set - - PowerPoint PPT Presentation

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The lattice of quasiorder lattices of algebras

• n a finite set

Danica Jakub´ ıkov´ a-Studenovsk´ a Reinhard P¨

• schel

S´ andor Radeleczki

P.J. ˇ Saf´ arik University Koˇ sice Technische Universit¨ at Dresden Miskolci Egyetem (University of Miskolc)

AAA88 Arbeitstagung Allgemeine Algebra Workshop on General Algebra Warsaw 20.6.2014

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (1/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Outline

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance simple

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (2/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Outline

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance simple

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (3/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

compatible quasiorders

A, F universal algebra compatible (invariant) relation q ⊆ A × A: For each f ∈ F (n-ary) we have f ⊲ q (f preserves q), i.e. (a1, b1), . . . , (an, bn) ∈ q = ⇒ (f (a1, . . . , an), f (b1, . . . , bn)) ∈ q . PordA, F compatible partial orders (refl., trans., antisymmetric) Generalization of PordA, F and ConA, F: QuordA, F compatible quasiorders (reflexive, transitive)

Remark

(QuordA, F, ⊆) is a lattice and it is a complete sublattice of the lattice (Quord(A), ⊆) of all quasiorders on A.

Problem

Describe the lattice L := ({QuordA, F | F set of operations on A}, ⊆).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (4/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

compatible quasiorders

A, F universal algebra compatible (invariant) relation q ⊆ A × A: For each f ∈ F (n-ary) we have f ⊲ q (f preserves q), i.e. (a1, b1), . . . , (an, bn) ∈ q = ⇒ (f (a1, . . . , an), f (b1, . . . , bn)) ∈ q . PordA, F compatible partial orders (refl., trans., antisymmetric) Generalization of PordA, F and ConA, F: QuordA, F compatible quasiorders (reflexive, transitive)

Remark

(QuordA, F, ⊆) is a lattice and it is a complete sublattice of the lattice (Quord(A), ⊆) of all quasiorders on A.

Problem

Describe the lattice L := ({QuordA, F | F set of operations on A}, ⊆).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (4/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

compatible quasiorders

A, F universal algebra compatible (invariant) relation q ⊆ A × A: For each f ∈ F (n-ary) we have f ⊲ q (f preserves q), i.e. (a1, b1), . . . , (an, bn) ∈ q = ⇒ (f (a1, . . . , an), f (b1, . . . , bn)) ∈ q . PordA, F compatible partial orders (refl., trans., antisymmetric) Generalization of PordA, F and ConA, F: QuordA, F compatible quasiorders (reflexive, transitive)

Remark

(QuordA, F, ⊆) is a lattice and it is a complete sublattice of the lattice (Quord(A), ⊆) of all quasiorders on A.

Problem

Describe the lattice L := ({QuordA, F | F set of operations on A}, ⊆).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (4/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

compatible quasiorders

A, F universal algebra compatible (invariant) relation q ⊆ A × A: For each f ∈ F (n-ary) we have f ⊲ q (f preserves q), i.e. (a1, b1), . . . , (an, bn) ∈ q = ⇒ (f (a1, . . . , an), f (b1, . . . , bn)) ∈ q . PordA, F compatible partial orders (refl., trans., antisymmetric) Generalization of PordA, F and ConA, F: QuordA, F compatible quasiorders (reflexive, transitive)

Remark

(QuordA, F, ⊆) is a lattice and it is a complete sublattice of the lattice (Quord(A), ⊆) of all quasiorders on A.

Problem

Describe the lattice L := ({QuordA, F | F set of operations on A}, ⊆).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (4/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

compatible quasiorders

A, F universal algebra compatible (invariant) relation q ⊆ A × A: For each f ∈ F (n-ary) we have f ⊲ q (f preserves q), i.e. (a1, b1), . . . , (an, bn) ∈ q = ⇒ (f (a1, . . . , an), f (b1, . . . , bn)) ∈ q . PordA, F compatible partial orders (refl., trans., antisymmetric) Generalization of PordA, F and ConA, F: QuordA, F compatible quasiorders (reflexive, transitive)

Remark

(QuordA, F, ⊆) is a lattice and it is a complete sublattice of the lattice (Quord(A), ⊆) of all quasiorders on A.

Problem

Describe the lattice L := ({QuordA, F | F set of operations on A}, ⊆).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (4/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

compatible quasiorders

A, F universal algebra compatible (invariant) relation q ⊆ A × A: For each f ∈ F (n-ary) we have f ⊲ q (f preserves q), i.e. (a1, b1), . . . , (an, bn) ∈ q = ⇒ (f (a1, . . . , an), f (b1, . . . , bn)) ∈ q . PordA, F compatible partial orders (refl., trans., antisymmetric) Generalization of PordA, F and ConA, F: QuordA, F compatible quasiorders (reflexive, transitive)

Remark

(QuordA, F, ⊆) is a lattice and it is a complete sublattice of the lattice (Quord(A), ⊆) of all quasiorders on A.

Problem

Describe the lattice L := ({QuordA, F | F set of operations on A}, ⊆).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (4/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

compatible quasiorders

A, F universal algebra compatible (invariant) relation q ⊆ A × A: For each f ∈ F (n-ary) we have f ⊲ q (f preserves q), i.e. (a1, b1), . . . , (an, bn) ∈ q = ⇒ (f (a1, . . . , an), f (b1, . . . , bn)) ∈ q . PordA, F compatible partial orders (refl., trans., antisymmetric) Generalization of PordA, F and ConA, F: QuordA, F compatible quasiorders (reflexive, transitive)

Remark

(QuordA, F, ⊆) is a lattice and it is a complete sublattice of the lattice (Quord(A), ⊆) of all quasiorders on A.

Problem

Describe the lattice L := ({QuordA, F | F set of operations on A}, ⊆).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (4/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Reduction to (mono)unary algebras

H := unary polynomial operations of A, F (i.e. H = F ∪ C(1)). Then (as for Con(A, F)) QuordA, F = QuordA, H QuordA, H =

• f ∈H

QuordA, f . Thus L = ({QuordA, H | H ≤ AA}, ⊆). Description of L: look for ∧- and ∨-irreducible elements Remark: End − Quord is a Galois connection (induced by ⊲).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (5/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Reduction to (mono)unary algebras

H := unary polynomial operations of A, F (i.e. H = F ∪ C(1)). Then (as for Con(A, F)) QuordA, F = QuordA, H QuordA, H =

• f ∈H

QuordA, f . Thus L = ({QuordA, H | H ≤ AA}, ⊆). Description of L: look for ∧- and ∨-irreducible elements Remark: End − Quord is a Galois connection (induced by ⊲).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (5/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Reduction to (mono)unary algebras

H := unary polynomial operations of A, F (i.e. H = F ∪ C(1)). Then (as for Con(A, F)) QuordA, F = QuordA, H QuordA, H =

• f ∈H

QuordA, f . Thus L = ({QuordA, H | H ≤ AA}, ⊆). Description of L: look for ∧- and ∨-irreducible elements Remark: End − Quord is a Galois connection (induced by ⊲).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (5/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Reduction to (mono)unary algebras

H := unary polynomial operations of A, F (i.e. H = F ∪ C(1)). Then (as for Con(A, F)) QuordA, F = QuordA, H QuordA, H =

• f ∈H

QuordA, f . Thus L = ({QuordA, H | H ≤ AA}, ⊆). Description of L: look for ∧- and ∨-irreducible elements Remark: End − Quord is a Galois connection (induced by ⊲).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (5/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Outline

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance simple

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (6/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

∨-irreducibles

q ∈ L := Quord(A, H) = ⇒ H ⊆ End q = ⇒ Quord(A, H) ⊇ Quord(A, End q) = ⇒ L =

q∈L Quord(A, End q).

Thus each ∨-irreducible element L = Quord(A, H) in L is of the form Lq = Quord(A, End q) for some q ∈ Quord(A). Question: Which q yield ∨-irreducibles? Answer: Every nontrivial q Proof: The quasiorder lattice Quord(A, End q) is a distributive lattice with at most six elements: Lq := Quord(A, End q) = q, q−1Quord(A) = {∆, q0, q, q−1, q ∨ q−1, ∇}.

(Result of P¨

• schel/Radeleczki 2007), see figure

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (7/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

∨-irreducibles

q ∈ L := Quord(A, H) = ⇒ H ⊆ End q = ⇒ Quord(A, H) ⊇ Quord(A, End q) = ⇒ L =

q∈L Quord(A, End q).

Thus each ∨-irreducible element L = Quord(A, H) in L is of the form Lq = Quord(A, End q) for some q ∈ Quord(A). Question: Which q yield ∨-irreducibles? Answer: Every nontrivial q Proof: The quasiorder lattice Quord(A, End q) is a distributive lattice with at most six elements: Lq := Quord(A, End q) = q, q−1Quord(A) = {∆, q0, q, q−1, q ∨ q−1, ∇}.

(Result of P¨

• schel/Radeleczki 2007), see figure

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (7/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

∨-irreducibles

q ∈ L := Quord(A, H) = ⇒ H ⊆ End q = ⇒ Quord(A, H) ⊇ Quord(A, End q) = ⇒ L =

q∈L Quord(A, End q).

Thus each ∨-irreducible element L = Quord(A, H) in L is of the form Lq = Quord(A, End q) for some q ∈ Quord(A). Question: Which q yield ∨-irreducibles? Answer: Every nontrivial q Proof: The quasiorder lattice Quord(A, End q) is a distributive lattice with at most six elements: Lq := Quord(A, End q) = q, q−1Quord(A) = {∆, q0, q, q−1, q ∨ q−1, ∇}.

(Result of P¨

• schel/Radeleczki 2007), see figure

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (7/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

∨-irreducibles

q ∈ L := Quord(A, H) = ⇒ H ⊆ End q = ⇒ Quord(A, H) ⊇ Quord(A, End q) = ⇒ L =

q∈L Quord(A, End q).

Thus each ∨-irreducible element L = Quord(A, H) in L is of the form Lq = Quord(A, End q) for some q ∈ Quord(A). Question: Which q yield ∨-irreducibles? Answer: Every nontrivial q Proof: The quasiorder lattice Quord(A, End q) is a distributive lattice with at most six elements: Lq := Quord(A, End q) = q, q−1Quord(A) = {∆, q0, q, q−1, q ∨ q−1, ∇}.

(Result of P¨

• schel/Radeleczki 2007), see figure

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (7/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

∨-irreducibles

q ∈ L := Quord(A, H) = ⇒ H ⊆ End q = ⇒ Quord(A, H) ⊇ Quord(A, End q) = ⇒ L =

q∈L Quord(A, End q).

Thus each ∨-irreducible element L = Quord(A, H) in L is of the form Lq = Quord(A, End q) for some q ∈ Quord(A). Question: Which q yield ∨-irreducibles? Answer: Every nontrivial q Proof: The quasiorder lattice Quord(A, End q) is a distributive lattice with at most six elements: Lq := Quord(A, End q) = q, q−1Quord(A) = {∆, q0, q, q−1, q ∨ q−1, ∇}.

(Result of P¨

• schel/Radeleczki 2007), see figure

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (7/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The quasiorder lattices Quord(A, End q), q ∈ Quord(A)

(b) ∇ ∆ q q−1 ∇ ∆ (a) q−1 q q ∨ q−1 q0 ∆ ∇ q = q−1 (c)

(a) q ∈ Quord(A) arbitrary quasiorder (general case) (b) q ∈ Pord(A) connected partial order (c) q ∈ Eq(A) equivalence relation

Corollary

The atoms in L are exactly those Quord(A, End q) where q ∈ Quord(A) \ {∆, ∇} is a connected partial order or an equivalence relation.

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (8/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The quasiorder lattices Quord(A, End q), q ∈ Quord(A)

(b) ∇ ∆ q q−1 ∇ ∆ (a) q−1 q q ∨ q−1 q0 ∆ ∇ q = q−1 (c)

(a) q ∈ Quord(A) arbitrary quasiorder (general case) (b) q ∈ Pord(A) connected partial order (c) q ∈ Eq(A) equivalence relation

Corollary

The atoms in L are exactly those Quord(A, End q) where q ∈ Quord(A) \ {∆, ∇} is a connected partial order or an equivalence relation.

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (8/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The quasiorder lattices Quord(A, End q), q ∈ Quord(A)

(b) ∇ ∆ q q−1 ∇ ∆ (a) q−1 q q ∨ q−1 q0 ∆ ∇ q = q−1 (c)

(a) q ∈ Quord(A) arbitrary quasiorder (general case) (b) q ∈ Pord(A) connected partial order (c) q ∈ Eq(A) equivalence relation

Corollary

The atoms in L are exactly those Quord(A, End q) where q ∈ Quord(A) \ {∆, ∇} is a connected partial order or an equivalence relation.

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (8/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The quasiorder lattices Quord(A, End q), q ∈ Quord(A)

(b) ∇ ∆ q q−1 ∇ ∆ (a) q−1 q q ∨ q−1 q0 ∆ ∇ q = q−1 (c)

(a) q ∈ Quord(A) arbitrary quasiorder (general case) (b) q ∈ Pord(A) connected partial order (c) q ∈ Eq(A) equivalence relation

Corollary

The atoms in L are exactly those Quord(A, End q) where q ∈ Quord(A) \ {∆, ∇} is a connected partial order or an equivalence relation.

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (8/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Join of atoms

1L := Quord(A) greatest element in L.

Proposition

There are two atoms in L whose join is 1L. More precisely, there are two linear orders λ1 and λ2 such that Lλ1 ∨ Lλ2 = Quord(A). Further, there are three equivalence relations θ1, θ2, θ3 such that Lθ1 ∨ Lθ2 ∨ Lθ3 = Quord(A). Proof Nozaki/Miyakawa/Pogosyan/Rosenberg (1995) showed that there are many (in particular at least two, say λ1 and λ2) pairwise “orthogonal” linear orders, what means that they together are preserved only by constants or the identity mapping. Thus Lλ1 ∨ Lλ2 = Quord End(Lλ1 ∪ Lλ2) ⊇ Quord End{λ1, λ2} = Quord(A). For equivalence relations analogous result by L. Zadori (1986).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (9/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Join of atoms

1L := Quord(A) greatest element in L.

Proposition

There are two atoms in L whose join is 1L. More precisely, there are two linear orders λ1 and λ2 such that Lλ1 ∨ Lλ2 = Quord(A). Further, there are three equivalence relations θ1, θ2, θ3 such that Lθ1 ∨ Lθ2 ∨ Lθ3 = Quord(A). Proof Nozaki/Miyakawa/Pogosyan/Rosenberg (1995) showed that there are many (in particular at least two, say λ1 and λ2) pairwise “orthogonal” linear orders, what means that they together are preserved only by constants or the identity mapping. Thus Lλ1 ∨ Lλ2 = Quord End(Lλ1 ∪ Lλ2) ⊇ Quord End{λ1, λ2} = Quord(A). For equivalence relations analogous result by L. Zadori (1986).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (9/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Join of atoms

1L := Quord(A) greatest element in L.

Proposition

There are two atoms in L whose join is 1L. More precisely, there are two linear orders λ1 and λ2 such that Lλ1 ∨ Lλ2 = Quord(A). Further, there are three equivalence relations θ1, θ2, θ3 such that Lθ1 ∨ Lθ2 ∨ Lθ3 = Quord(A). Proof Nozaki/Miyakawa/Pogosyan/Rosenberg (1995) showed that there are many (in particular at least two, say λ1 and λ2) pairwise “orthogonal” linear orders, what means that they together are preserved only by constants or the identity mapping. Thus Lλ1 ∨ Lλ2 = Quord End(Lλ1 ∪ Lλ2) ⊇ Quord End{λ1, λ2} = Quord(A). For equivalence relations analogous result by L. Zadori (1986).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (9/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Join of atoms

1L := Quord(A) greatest element in L.

Proposition

There are two atoms in L whose join is 1L. More precisely, there are two linear orders λ1 and λ2 such that Lλ1 ∨ Lλ2 = Quord(A). Further, there are three equivalence relations θ1, θ2, θ3 such that Lθ1 ∨ Lθ2 ∨ Lθ3 = Quord(A). Proof Nozaki/Miyakawa/Pogosyan/Rosenberg (1995) showed that there are many (in particular at least two, say λ1 and λ2) pairwise “orthogonal” linear orders, what means that they together are preserved only by constants or the identity mapping. Thus Lλ1 ∨ Lλ2 = Quord End(Lλ1 ∪ Lλ2) ⊇ Quord End{λ1, λ2} = Quord(A). For equivalence relations analogous result by L. Zadori (1986).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (9/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Join of atoms

1L := Quord(A) greatest element in L.

Proposition

There are two atoms in L whose join is 1L. More precisely, there are two linear orders λ1 and λ2 such that Lλ1 ∨ Lλ2 = Quord(A). Further, there are three equivalence relations θ1, θ2, θ3 such that Lθ1 ∨ Lθ2 ∨ Lθ3 = Quord(A). Proof Nozaki/Miyakawa/Pogosyan/Rosenberg (1995) showed that there are many (in particular at least two, say λ1 and λ2) pairwise “orthogonal” linear orders, what means that they together are preserved only by constants or the identity mapping. Thus Lλ1 ∨ Lλ2 = Quord End(Lλ1 ∪ Lλ2) ⊇ Quord End{λ1, λ2} = Quord(A). For equivalence relations analogous result by L. Zadori (1986).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (9/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Outline

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance simple

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (10/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

∧-irreducibles and coatoms

∧-irreducibles are of the form Quord(A, f ) for some nontrivial f ∈ AA known only for special types of f (permutations, acyclic mappings) coatoms are known completely How many coatoms are needed for trivial intersection?

Proposition

There are two or three coatoms in L whose meet is 0L. More precisely, for |A| > 5, there are two coatoms Quord(A, f ) and Quord(A, g) such that Quord(A, g) ∩ Quord(A, h) = {∆, ∇}. For |A| ≤ 5, three coatoms are necessary (and sufficient) for this property.

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (11/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

∧-irreducibles and coatoms

∧-irreducibles are of the form Quord(A, f ) for some nontrivial f ∈ AA known only for special types of f (permutations, acyclic mappings) coatoms are known completely How many coatoms are needed for trivial intersection?

Proposition

There are two or three coatoms in L whose meet is 0L. More precisely, for |A| > 5, there are two coatoms Quord(A, f ) and Quord(A, g) such that Quord(A, g) ∩ Quord(A, h) = {∆, ∇}. For |A| ≤ 5, three coatoms are necessary (and sufficient) for this property.

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (11/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

∧-irreducibles and coatoms

∧-irreducibles are of the form Quord(A, f ) for some nontrivial f ∈ AA known only for special types of f (permutations, acyclic mappings) coatoms are known completely How many coatoms are needed for trivial intersection?

Proposition

There are two or three coatoms in L whose meet is 0L. More precisely, for |A| > 5, there are two coatoms Quord(A, f ) and Quord(A, g) such that Quord(A, g) ∩ Quord(A, h) = {∆, ∇}. For |A| ≤ 5, three coatoms are necessary (and sufficient) for this property.

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (11/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

∧-irreducibles and coatoms

∧-irreducibles are of the form Quord(A, f ) for some nontrivial f ∈ AA known only for special types of f (permutations, acyclic mappings) coatoms are known completely How many coatoms are needed for trivial intersection?

Proposition

There are two or three coatoms in L whose meet is 0L. More precisely, for |A| > 5, there are two coatoms Quord(A, f ) and Quord(A, g) such that Quord(A, g) ∩ Quord(A, h) = {∆, ∇}. For |A| ≤ 5, three coatoms are necessary (and sufficient) for this property.

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (11/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

∧-irreducibles and coatoms

∧-irreducibles are of the form Quord(A, f ) for some nontrivial f ∈ AA known only for special types of f (permutations, acyclic mappings) coatoms are known completely How many coatoms are needed for trivial intersection?

Proposition

There are two or three coatoms in L whose meet is 0L. More precisely, for |A| > 5, there are two coatoms Quord(A, f ) and Quord(A, g) such that Quord(A, g) ∩ Quord(A, h) = {∆, ∇}. For |A| ≤ 5, three coatoms are necessary (and sufficient) for this property.

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (11/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Outline

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance simple

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (12/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Tolerance relations

tolerance relation: reflexive and symmetric binary relation on a set. For a lattice (V , ∧, ∨): Tol(V ) - set of all compatible tolerance relations ̺ on L With respect to set-theoretic inclusion the tolerances form an algebraic lattice (Tol(V ), ∩, ⊔): least element ∆V := {(x, x) | x ∈ V } greatest element ∇V := V × V (called trivial tolerance relations). (V , ∧, ∨) tolerance simple : ⇐ ⇒ no nontrivial tolerances, i.e., Tol(V ) = {∆V , ∇V }. Remark: finite tolerance simple lattices are order-polynomially complete

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (13/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Tolerance relations

tolerance relation: reflexive and symmetric binary relation on a set. For a lattice (V , ∧, ∨): Tol(V ) - set of all compatible tolerance relations ̺ on L With respect to set-theoretic inclusion the tolerances form an algebraic lattice (Tol(V ), ∩, ⊔): least element ∆V := {(x, x) | x ∈ V } greatest element ∇V := V × V (called trivial tolerance relations). (V , ∧, ∨) tolerance simple : ⇐ ⇒ no nontrivial tolerances, i.e., Tol(V ) = {∆V , ∇V }. Remark: finite tolerance simple lattices are order-polynomially complete

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (13/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Tolerance relations

tolerance relation: reflexive and symmetric binary relation on a set. For a lattice (V , ∧, ∨): Tol(V ) - set of all compatible tolerance relations ̺ on L With respect to set-theoretic inclusion the tolerances form an algebraic lattice (Tol(V ), ∩, ⊔): least element ∆V := {(x, x) | x ∈ V } greatest element ∇V := V × V (called trivial tolerance relations). (V , ∧, ∨) tolerance simple : ⇐ ⇒ no nontrivial tolerances, i.e., Tol(V ) = {∆V , ∇V }. Remark: finite tolerance simple lattices are order-polynomially complete

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (13/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Tolerance relations

tolerance relation: reflexive and symmetric binary relation on a set. For a lattice (V , ∧, ∨): Tol(V ) - set of all compatible tolerance relations ̺ on L With respect to set-theoretic inclusion the tolerances form an algebraic lattice (Tol(V ), ∩, ⊔): least element ∆V := {(x, x) | x ∈ V } greatest element ∇V := V × V (called trivial tolerance relations). (V , ∧, ∨) tolerance simple : ⇐ ⇒ no nontrivial tolerances, i.e., Tol(V ) = {∆V , ∇V }. Remark: finite tolerance simple lattices are order-polynomially complete

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (13/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Tolerance relations

tolerance relation: reflexive and symmetric binary relation on a set. For a lattice (V , ∧, ∨): Tol(V ) - set of all compatible tolerance relations ̺ on L With respect to set-theoretic inclusion the tolerances form an algebraic lattice (Tol(V ), ∩, ⊔): least element ∆V := {(x, x) | x ∈ V } greatest element ∇V := V × V (called trivial tolerance relations). (V , ∧, ∨) tolerance simple : ⇐ ⇒ no nontrivial tolerances, i.e., Tol(V ) = {∆V , ∇V }. Remark: finite tolerance simple lattices are order-polynomially complete

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (13/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Tolerance relations

tolerance relation: reflexive and symmetric binary relation on a set. For a lattice (V , ∧, ∨): Tol(V ) - set of all compatible tolerance relations ̺ on L With respect to set-theoretic inclusion the tolerances form an algebraic lattice (Tol(V ), ∩, ⊔): least element ∆V := {(x, x) | x ∈ V } greatest element ∇V := V × V (called trivial tolerance relations). (V , ∧, ∨) tolerance simple : ⇐ ⇒ no nontrivial tolerances, i.e., Tol(V ) = {∆V , ∇V }. Remark: finite tolerance simple lattices are order-polynomially complete

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (13/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Properties of Tol(V )

For T ∈ Tol(V ), T =

(x,y)∈T T(x, y) where

T(x, y) - least tolerance in Tol(L) containing (x, y) ∈ V 2 We have T(x ∧ y, y) = T(x, x ∨ y) (cf. Figure), (1) T(x′, y′) ⊆ T(x, y) whenever x ≤ x′ ≤ y′ ≤ y, (2) (0V , 1V ) ∈ T ∈ Tol(V ) = ⇒ T = ∇V . (3)

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (14/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Properties of Tol(V )

For T ∈ Tol(V ), T =

(x,y)∈T T(x, y) where

T(x, y) - least tolerance in Tol(L) containing (x, y) ∈ V 2 We have T(x ∧ y, y) = T(x, x ∨ y) (cf. Figure), (1) T(x′, y′) ⊆ T(x, y) whenever x ≤ x′ ≤ y′ ≤ y, (2) (0V , 1V ) ∈ T ∈ Tol(V ) = ⇒ T = ∇V . (3)

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (14/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Properties of Tol(V )

For T ∈ Tol(V ), T =

(x,y)∈T T(x, y) where

T(x, y) - least tolerance in Tol(L) containing (x, y) ∈ V 2 We have T(x ∧ y, y) = T(x, x ∨ y) (cf. Figure), (1) T(x′, y′) ⊆ T(x, y) whenever x ≤ x′ ≤ y′ ≤ y, (2) (0V , 1V ) ∈ T ∈ Tol(V ) = ⇒ T = ∇V . (3)

(1) T(x, x ∨ y) = T(x ∧ y, y) x ∧ y x y x ∨ y

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (14/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Tolerance simplicity

∨-irreducible p ∈ V , = ⇒ ∃ unique p∗ ≺ p

Proposition

Let (V , ∧, ∨) be an arbitrary finite lattice and let T(p∗, p) = ∇V for every ∨-irreducible element p ∈ V . Then V is tolerance simple. Sketch of the proof. Let α be an atom in Tol(V ) known: Then α = T(a, b) for some a, b ∈ V and a ≺ b. V finite = ⇒ ∃ ∨-irreducible p ∈ V : p ≤ b, p ≤ a, but p∗ ≤ a (this can be proved via elements p of minimal height in V ). Hence a ∨ p = b (since a ≺ b) and a ∧ p = p∗ (since p∗ ≺ p). Apply (1): α = T(a, b) = T(a, a ∨ p) = T(a ∧ p, p) = T(p∗, p) = ∇V . Thus ∇V is the only atom of Tol(V ), i.e., Tol(V ) = {∆V , ∇V }, V is tolerance simple.

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (15/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Tolerance simplicity

∨-irreducible p ∈ V , = ⇒ ∃ unique p∗ ≺ p

Proposition

Let (V , ∧, ∨) be an arbitrary finite lattice and let T(p∗, p) = ∇V for every ∨-irreducible element p ∈ V . Then V is tolerance simple. Sketch of the proof. Let α be an atom in Tol(V ) known: Then α = T(a, b) for some a, b ∈ V and a ≺ b. V finite = ⇒ ∃ ∨-irreducible p ∈ V : p ≤ b, p ≤ a, but p∗ ≤ a (this can be proved via elements p of minimal height in V ). Hence a ∨ p = b (since a ≺ b) and a ∧ p = p∗ (since p∗ ≺ p). Apply (1): α = T(a, b) = T(a, a ∨ p) = T(a ∧ p, p) = T(p∗, p) = ∇V . Thus ∇V is the only atom of Tol(V ), i.e., Tol(V ) = {∆V , ∇V }, V is tolerance simple.

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (15/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Tolerance simplicity

∨-irreducible p ∈ V , = ⇒ ∃ unique p∗ ≺ p

Proposition

Let (V , ∧, ∨) be an arbitrary finite lattice and let T(p∗, p) = ∇V for every ∨-irreducible element p ∈ V . Then V is tolerance simple. Sketch of the proof. Let α be an atom in Tol(V ) known: Then α = T(a, b) for some a, b ∈ V and a ≺ b. V finite = ⇒ ∃ ∨-irreducible p ∈ V : p ≤ b, p ≤ a, but p∗ ≤ a (this can be proved via elements p of minimal height in V ). Hence a ∨ p = b (since a ≺ b) and a ∧ p = p∗ (since p∗ ≺ p). Apply (1): α = T(a, b) = T(a, a ∨ p) = T(a ∧ p, p) = T(p∗, p) = ∇V . Thus ∇V is the only atom of Tol(V ), i.e., Tol(V ) = {∆V , ∇V }, V is tolerance simple.

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (15/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The lattice L is tolerance simple

Theorem

For |A| ≥ 4, the lattice L is tolerance simple.

• Proof. By Proposition: it is sufficient to prove T(L∗, L) = ∇L for

every ∨-irreducible element L ∈ L. Part 1: Show T(L∗, L) = ∇L for every atom L ∈ L (clearly L∗ = 0L for atoms L). Thus, in particular, we have T(0L, Lλ) = ∇L for every linear order λ ∈ Lord(A) (because Lλ := {∆, λ, λ−1, ∇} = Quord(A, End λ) is an atom). Part 2: It remains to consider the ∨-irreducibles L which are not

• atoms. By our result, they are of the form L = Lq for some

non-symmetric q. One can show that, for each non-symmetric q ∈ Quord(A), there exist K ∈ L and λ ∈ Lord(A), such that K ∩ Lq = L∗

q, K ∩ Lλ = 0L and K ∨ Lλ ≤ K ∨ Lq. With properties

(1), (2) and Part 1, we get ∇L = T(0L, Lλ) = T(K ∩ Lλ, Lλ) = T(K, K ∨ Lλ) ⊆ T(K, K ∨ Lq) = T(K ∩ Lq, Lq) = T(L∗

q, Lq)

(see Figure).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (16/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The lattice L is tolerance simple

Theorem

For |A| ≥ 4, the lattice L is tolerance simple.

• Proof. By Proposition: it is sufficient to prove T(L∗, L) = ∇L for

every ∨-irreducible element L ∈ L. Part 1: Show T(L∗, L) = ∇L for every atom L ∈ L (clearly L∗ = 0L for atoms L). Thus, in particular, we have T(0L, Lλ) = ∇L for every linear order λ ∈ Lord(A) (because Lλ := {∆, λ, λ−1, ∇} = Quord(A, End λ) is an atom). Part 2: It remains to consider the ∨-irreducibles L which are not

• atoms. By our result, they are of the form L = Lq for some

non-symmetric q. One can show that, for each non-symmetric q ∈ Quord(A), there exist K ∈ L and λ ∈ Lord(A), such that K ∩ Lq = L∗

q, K ∩ Lλ = 0L and K ∨ Lλ ≤ K ∨ Lq. With properties

(1), (2) and Part 1, we get ∇L = T(0L, Lλ) = T(K ∩ Lλ, Lλ) = T(K, K ∨ Lλ) ⊆ T(K, K ∨ Lq) = T(K ∩ Lq, Lq) = T(L∗

q, Lq)

(see Figure).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (16/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The lattice L is tolerance simple

Theorem

For |A| ≥ 4, the lattice L is tolerance simple.

• Proof. By Proposition: it is sufficient to prove T(L∗, L) = ∇L for

every ∨-irreducible element L ∈ L. Part 1: Show T(L∗, L) = ∇L for every atom L ∈ L (clearly L∗ = 0L for atoms L). Thus, in particular, we have T(0L, Lλ) = ∇L for every linear order λ ∈ Lord(A) (because Lλ := {∆, λ, λ−1, ∇} = Quord(A, End λ) is an atom). Part 2: It remains to consider the ∨-irreducibles L which are not

• atoms. By our result, they are of the form L = Lq for some

non-symmetric q. One can show that, for each non-symmetric q ∈ Quord(A), there exist K ∈ L and λ ∈ Lord(A), such that K ∩ Lq = L∗

q, K ∩ Lλ = 0L and K ∨ Lλ ≤ K ∨ Lq. With properties

(1), (2) and Part 1, we get ∇L = T(0L, Lλ) = T(K ∩ Lλ, Lλ) = T(K, K ∨ Lλ) ⊆ T(K, K ∨ Lq) = T(K ∩ Lq, Lq) = T(L∗

q, Lq)

(see Figure).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (16/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The lattice L is tolerance simple

Theorem

For |A| ≥ 4, the lattice L is tolerance simple.

• Proof. By Proposition: it is sufficient to prove T(L∗, L) = ∇L for

every ∨-irreducible element L ∈ L. Part 1: Show T(L∗, L) = ∇L for every atom L ∈ L (clearly L∗ = 0L for atoms L). Thus, in particular, we have T(0L, Lλ) = ∇L for every linear order λ ∈ Lord(A) (because Lλ := {∆, λ, λ−1, ∇} = Quord(A, End λ) is an atom). Part 2: It remains to consider the ∨-irreducibles L which are not

• atoms. By our result, they are of the form L = Lq for some

non-symmetric q. One can show that, for each non-symmetric q ∈ Quord(A), there exist K ∈ L and λ ∈ Lord(A), such that K ∩ Lq = L∗

q, K ∩ Lλ = 0L and K ∨ Lλ ≤ K ∨ Lq. With properties

(1), (2) and Part 1, we get ∇L = T(0L, Lλ) = T(K ∩ Lλ, Lλ) = T(K, K ∨ Lλ) ⊆ T(K, K ∨ Lq) = T(K ∩ Lq, Lq) = T(L∗

q, Lq)

(see Figure).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (16/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The lattice L is tolerance simple

Theorem

For |A| ≥ 4, the lattice L is tolerance simple.

• Proof. By Proposition: it is sufficient to prove T(L∗, L) = ∇L for

every ∨-irreducible element L ∈ L. Part 1: Show T(L∗, L) = ∇L for every atom L ∈ L (clearly L∗ = 0L for atoms L). Thus, in particular, we have T(0L, Lλ) = ∇L for every linear order λ ∈ Lord(A) (because Lλ := {∆, λ, λ−1, ∇} = Quord(A, End λ) is an atom). Part 2: It remains to consider the ∨-irreducibles L which are not

• atoms. By our result, they are of the form L = Lq for some

non-symmetric q. One can show that, for each non-symmetric q ∈ Quord(A), there exist K ∈ L and λ ∈ Lord(A), such that K ∩ Lq = L∗

q, K ∩ Lλ = 0L and K ∨ Lλ ≤ K ∨ Lq. With properties

(1), (2) and Part 1, we get ∇L = T(0L, Lλ) = T(K ∩ Lλ, Lλ) = T(K, K ∨ Lλ) ⊆ T(K, K ∨ Lq) = T(K ∩ Lq, Lq) = T(L∗

q, Lq)

(see Figure).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (16/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The lattice L is tolerance simple

Theorem

For |A| ≥ 4, the lattice L is tolerance simple.

• Proof. By Proposition: it is sufficient to prove T(L∗, L) = ∇L for

every ∨-irreducible element L ∈ L. Part 1: Show T(L∗, L) = ∇L for every atom L ∈ L (clearly L∗ = 0L for atoms L). Thus, in particular, we have T(0L, Lλ) = ∇L for every linear order λ ∈ Lord(A) (because Lλ := {∆, λ, λ−1, ∇} = Quord(A, End λ) is an atom). Part 2: It remains to consider the ∨-irreducibles L which are not

• atoms. By our result, they are of the form L = Lq for some

non-symmetric q. One can show that, for each non-symmetric q ∈ Quord(A), there exist K ∈ L and λ ∈ Lord(A), such that K ∩ Lq = L∗

q, K ∩ Lλ = 0L and K ∨ Lλ ≤ K ∨ Lq. With properties

(1), (2) and Part 1, we get ∇L = T(0L, Lλ) = T(K ∩ Lλ, Lλ) = T(K, K ∨ Lλ) ⊆ T(K, K ∨ Lq) = T(K ∩ Lq, Lq) = T(L∗

q, Lq)

(see Figure).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (16/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The lattice L is tolerance simple

Theorem

For |A| ≥ 4, the lattice L is tolerance simple.

• Proof. By Proposition: it is sufficient to prove T(L∗, L) = ∇L for

every ∨-irreducible element L ∈ L. Part 1: Show T(L∗, L) = ∇L for every atom L ∈ L (clearly L∗ = 0L for atoms L). Thus, in particular, we have T(0L, Lλ) = ∇L for every linear order λ ∈ Lord(A) (because Lλ := {∆, λ, λ−1, ∇} = Quord(A, End λ) is an atom). Part 2: It remains to consider the ∨-irreducibles L which are not

• atoms. By our result, they are of the form L = Lq for some

non-symmetric q. One can show that, for each non-symmetric q ∈ Quord(A), there exist K ∈ L and λ ∈ Lord(A), such that K ∩ Lq = L∗

q, K ∩ Lλ = 0L and K ∨ Lλ ≤ K ∨ Lq. With properties

(1), (2) and Part 1, we get ∇L = T(0L, Lλ) = T(K ∩ Lλ, Lλ) = T(K, K ∨ Lλ) ⊆ T(K, K ∨ Lq) = T(K ∩ Lq, Lq) = T(L∗

q, Lq)

(see Figure).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (16/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The lattice L is tolerance simple

Theorem

For |A| ≥ 4, the lattice L is tolerance simple.

• Proof. By Proposition: it is sufficient to prove T(L∗, L) = ∇L for

every ∨-irreducible element L ∈ L. Part 1: Show T(L∗, L) = ∇L for every atom L ∈ L (clearly L∗ = 0L for atoms L). Thus, in particular, we have T(0L, Lλ) = ∇L for every linear order λ ∈ Lord(A) (because Lλ := {∆, λ, λ−1, ∇} = Quord(A, End λ) is an atom). Part 2: It remains to consider the ∨-irreducibles L which are not

• atoms. By our result, they are of the form L = Lq for some

non-symmetric q. One can show that, for each non-symmetric q ∈ Quord(A), there exist K ∈ L and λ ∈ Lord(A), such that K ∩ Lq = L∗

q, K ∩ Lλ = 0L and K ∨ Lλ ≤ K ∨ Lq. With properties

(1), (2) and Part 1, we get ∇L = T(0L, Lλ) = T(K ∩ Lλ, Lλ) = T(K, K ∨ Lλ) ⊆ T(K, K ∨ Lq) = T(K ∩ Lq, Lq) = T(L∗

q, Lq)

(see Figure).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (16/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

The lattice L is tolerance simple

Theorem

For |A| ≥ 4, the lattice L is tolerance simple.

• Proof. By Proposition: it is sufficient to prove T(L∗, L) = ∇L for

every ∨-irreducible element L ∈ L. Part 1: Show T(L∗, L) = ∇L for every atom L ∈ L (clearly L∗ = 0L for atoms L). Thus, in particular, we have T(0L, Lλ) = ∇L for every linear order λ ∈ Lord(A) (because Lλ := {∆, λ, λ−1, ∇} = Quord(A, End λ) is an atom). Part 2: It remains to consider the ∨-irreducibles L which are not

• atoms. By our result, they are of the form L = Lq for some

non-symmetric q. One can show that, for each non-symmetric q ∈ Quord(A), there exist K ∈ L and λ ∈ Lord(A), such that K ∩ Lq = L∗

q, K ∩ Lλ = 0L and K ∨ Lλ ≤ K ∨ Lq. With properties

(1), (2) and Part 1, we get ∇L = T(0L, Lλ) = T(K ∩ Lλ, Lλ) = T(K, K ∨ Lλ) ⊆ T(K, K ∨ Lq) = T(K ∩ Lq, Lq) = T(L∗

q, Lq)

(see Figure).

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (16/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

K ∨ Lq Lλ ∨ K K = Quord(A, f ) Lq L∗

q

Lλ 0L (1) T(x, x ∨ y) = T(x ∧ y, y) x ∧ y x y x ∨ y ∇L ∇L ∇L ∇L

Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (17/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance

Other properties of L (around modularity)

Proposition

For |A| ≥ 4, the lattice L has neither of the following properties: 0-modular, 1-modular, upper semimodular, lower semimodular1.

1L is not lower semimodular even for |A| = 3. Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (18/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (19/20)

Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolerance Warsaw, June, 2014,

• R. P¨
• schel, The lattice of quasiorder lattices (20/20)