Notions and notations ∨-irreducibles (in particular atoms) of L ∧-irreducibles (in particular coatoms) of L The lattice L is tolera
The lattice of quasiorder lattices of algebras
on a finite set
Danica Jakubı́ková-Studenovská
Reinhard Pöschel
Sándor Radeleczki
P.J. Šafárik University Košice
Technische Universität Dresden
Miskolci Egyetem (University of Miskolc)
AAA88
Arbeitstagung Allgemeine Algebra
Workshop on General Algebra
Warsaw 20.6.2014
Warsaw, June, 2014,
R. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
compatible quasiorders
hA, F i 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 .
PordhA, F i compatible partial orders (refl., trans., antisymmetric)
Generalization of PordhA, F i and ConhA, F i:
QuordhA, F i compatible quasiorders (reflexive, transitive)
Remark
(QuordhA, F i, ⊆) is a lattice and it is a complete sublattice of the
lattice (Quord(A), ⊆) of all quasiorders on A.
Problem
Describe the lattice
L := ({QuordhA, F i | F set of operations on A}, ⊆).
Warsaw, June, 2014,
R. Pöschel, 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 tolera
compatible quasiorders
hA, F i 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 .
PordhA, F i compatible partial orders (refl., trans., antisymmetric)
Generalization of PordhA, F i and ConhA, F i:
QuordhA, F i compatible quasiorders (reflexive, transitive)
Remark
(QuordhA, F i, ⊆) is a lattice and it is a complete sublattice of the
lattice (Quord(A), ⊆) of all quasiorders on A.
Problem
Describe the lattice
L := ({QuordhA, F i | F set of operations on A}, ⊆).
Warsaw, June, 2014,
R. Pöschel, 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 tolera
compatible quasiorders
hA, F i 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 .
PordhA, F i compatible partial orders (refl., trans., antisymmetric)
Generalization of PordhA, F i and ConhA, F i:
QuordhA, F i compatible quasiorders (reflexive, transitive)
Remark
(QuordhA, F i, ⊆) is a lattice and it is a complete sublattice of the
lattice (Quord(A), ⊆) of all quasiorders on A.
Problem
Describe the lattice
L := ({QuordhA, F i | F set of operations on A}, ⊆).
Warsaw, June, 2014,
R. Pöschel, 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 tolera
compatible quasiorders
hA, F i 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 .
PordhA, F i compatible partial orders (refl., trans., antisymmetric)
Generalization of PordhA, F i and ConhA, F i:
QuordhA, F i compatible quasiorders (reflexive, transitive)
Remark
(QuordhA, F i, ⊆) is a lattice and it is a complete sublattice of the
lattice (Quord(A), ⊆) of all quasiorders on A.
Problem
Describe the lattice
L := ({QuordhA, F i | F set of operations on A}, ⊆).
Warsaw, June, 2014,
R. Pöschel, 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 tolera
compatible quasiorders
hA, F i 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 .
PordhA, F i compatible partial orders (refl., trans., antisymmetric)
Generalization of PordhA, F i and ConhA, F i:
QuordhA, F i compatible quasiorders (reflexive, transitive)
Remark
(QuordhA, F i, ⊆) is a lattice and it is a complete sublattice of the
lattice (Quord(A), ⊆) of all quasiorders on A.
Problem
Describe the lattice
L := ({QuordhA, F i | F set of operations on A}, ⊆).
Warsaw, June, 2014,
R. Pöschel, 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 tolera
compatible quasiorders
hA, F i 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 .
PordhA, F i compatible partial orders (refl., trans., antisymmetric)
Generalization of PordhA, F i and ConhA, F i:
QuordhA, F i compatible quasiorders (reflexive, transitive)
Remark
(QuordhA, F i, ⊆) is a lattice and it is a complete sublattice of the
lattice (Quord(A), ⊆) of all quasiorders on A.
Problem
Describe the lattice
L := ({QuordhA, F i | F set of operations on A}, ⊆).
Warsaw, June, 2014,
R. Pöschel, 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 tolera
compatible quasiorders
hA, F i 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 .
PordhA, F i compatible partial orders (refl., trans., antisymmetric)
Generalization of PordhA, F i and ConhA, F i:
QuordhA, F i compatible quasiorders (reflexive, transitive)
Remark
(QuordhA, F i, ⊆) is a lattice and it is a complete sublattice of the
lattice (Quord(A), ⊆) of all quasiorders on A.
Problem
Describe the lattice
L := ({QuordhA, F i | F set of operations on A}, ⊆).
Warsaw, June, 2014,
R. Pöschel, 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 tolera
Reduction to (mono)unary algebras
H := unary polynomial operations of hA, F i (i.e. H = hF ∪ C i(1) ).
Then (as for Con(A, F ))
QuordhA, F i = QuordhA, Hi
\
QuordhA, Hi =
QuordhA, f i.
f ∈H
Thus L = ({QuordhA, Hi | H ≤ AA }, ⊆).
Description of L: look for ∧- and ∨-irreducible elements
Remark: End − Quord is a Galois connection (induced by .).
Warsaw, June, 2014,
R. Pöschel, 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 tolera
Reduction to (mono)unary algebras
H := unary polynomial operations of hA, F i (i.e. H = hF ∪ C i(1) ).
Then (as for Con(A, F ))
QuordhA, F i = QuordhA, Hi
\
QuordhA, Hi =
QuordhA, f i.
f ∈H
Thus L = ({QuordhA, Hi | H ≤ AA }, ⊆).
Description of L: look for ∧- and ∨-irreducible elements
Remark: End − Quord is a Galois connection (induced by .).
Warsaw, June, 2014,
R. Pöschel, 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 tolera
Reduction to (mono)unary algebras
H := unary polynomial operations of hA, F i (i.e. H = hF ∪ C i(1) ).
Then (as for Con(A, F ))
QuordhA, F i = QuordhA, Hi
\
QuordhA, Hi =
QuordhA, f i.
f ∈H
Thus L = ({QuordhA, Hi | H ≤ AA }, ⊆).
Description of L: look for ∧- and ∨-irreducible elements
Remark: End − Quord is a Galois connection (induced by .).
Warsaw, June, 2014,
R. Pöschel, 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 tolera
Reduction to (mono)unary algebras
H := unary polynomial operations of hA, F i (i.e. H = hF ∪ C i(1) ).
Then (as for Con(A, F ))
QuordhA, F i = QuordhA, Hi
\
QuordhA, Hi =
QuordhA, f i.
f ∈H
Thus L = ({QuordhA, Hi | H ≤ AA }, ⊆).
Description of L: look for ∧- and ∨-irreducible elements
Remark: End − Quord is a Galois connection (induced by .).
Warsaw, June, 2014,
R. Pöschel, 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 tolera
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. Pöschel, 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 tolera
∨-irreducibles
q ∈ L := Quord(A, H) =⇒ H ⊆ End q =⇒
S
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) = hq, q −1 iQuord(A) = {∆, q0 , q, q −1 , q ∨ q −1 , ∇}.
(Result of Pöschel/Radeleczki 2007), see figure
Warsaw, June, 2014,
R. Pöschel, 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 tolera
∨-irreducibles
q ∈ L := Quord(A, H) =⇒ H ⊆ End q =⇒
S
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) = hq, q −1 iQuord(A) = {∆, q0 , q, q −1 , q ∨ q −1 , ∇}.
(Result of Pöschel/Radeleczki 2007), see figure
Warsaw, June, 2014,
R. Pöschel, 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 tolera
∨-irreducibles
q ∈ L := Quord(A, H) =⇒ H ⊆ End q =⇒
S
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) = hq, q −1 iQuord(A) = {∆, q0 , q, q −1 , q ∨ q −1 , ∇}.
(Result of Pöschel/Radeleczki 2007), see figure
Warsaw, June, 2014,
R. Pöschel, 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 tolera
∨-irreducibles
q ∈ L := Quord(A, H) =⇒ H ⊆ End q =⇒
S
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) = hq, q −1 iQuord(A) = {∆, q0 , q, q −1 , q ∨ q −1 , ∇}.
(Result of Pöschel/Radeleczki 2007), see figure
Warsaw, June, 2014,
R. Pöschel, 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 tolera
∨-irreducibles
q ∈ L := Quord(A, H) =⇒ H ⊆ End q =⇒
S
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) = hq, q −1 iQuord(A) = {∆, q0 , q, q −1 , q ∨ q −1 , ∇}.
(Result of Pöschel/Radeleczki 2007), see figure
Warsaw, June, 2014,
R. Pöschel, 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 tolera
The quasiorder lattices Quord(A, End q), q ∈ Quord(A)
∇
∇
q ∨ q −1
q −1
q
q0
∇
q −1
q
∆
q = q −1
∆
∆
(a)
(b)
(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. Pöschel, 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 tolera
The quasiorder lattices Quord(A, End q), q ∈ Quord(A)
∇
∇
q ∨ q −1
q −1
q
q0
∇
q −1
q
∆
q = q −1
∆
∆
(a)
(b)
(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. Pöschel, 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 tolera
The quasiorder lattices Quord(A, End q), q ∈ Quord(A)
∇
∇
q ∨ q −1
q −1
q
q0
∇
q −1
q
∆
q = q −1
∆
∆
(a)
(b)
(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. Pöschel, 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 tolera
The quasiorder lattices Quord(A, End q), q ∈ Quord(A)
∇
∇
q ∨ q −1
q −1
q
q0
∇
q −1
q
∆
q = q −1
∆
∆
(a)
(b)
(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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
∧-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. Pöschel, 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 tolera
∧-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. Pöschel, 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 tolera
∧-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. Pöschel, 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 tolera
∧-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. Pöschel, 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 tolera
∧-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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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 ), ∩, t):
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. Pöschel, 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 tolera
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 ), ∩, t):
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. Pöschel, 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 tolera
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 ), ∩, t):
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. Pöschel, 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 tolera
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 ), ∩, t):
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. Pöschel, 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 tolera
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 ), ∩, t):
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. Pöschel, 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 tolera
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 ), ∩, t):
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. Pöschel, 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 tolera
Properties of Tol(V )
F
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 )
0
Warsaw, June, 2014,
0
(cf. Figure),
0
(1)
0
T (x , y ) ⊆ T (x, y ) whenever x ≤ x ≤ y ≤ y ,
(2)
(0V , 1V ) ∈ T ∈ Tol(V ) =⇒ T = ∇V .
(3)
R. Pöschel, 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 tolera
Properties of Tol(V )
F
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 )
0
Warsaw, June, 2014,
0
(cf. Figure),
0
(1)
0
T (x , y ) ⊆ T (x, y ) whenever x ≤ x ≤ y ≤ y ,
(2)
(0V , 1V ) ∈ T ∈ Tol(V ) =⇒ T = ∇V .
(3)
R. Pöschel, 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 tolera
Properties of Tol(V )
F
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 )
0
(cf. Figure),
0
0
(1)
0
T (x , y ) ⊆ T (x, y ) whenever x ≤ x ≤ y ≤ y ,
(2)
(0V , 1V ) ∈ T ∈ Tol(V ) =⇒ T = ∇V .
(3)
x ∨y
y
x
x ∧y
(1)
T (x, x ∨ y ) = T (x ∧ y , y )
Warsaw, June, 2014,
R. Pöschel, 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 tolera
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 6≤ 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. Pöschel, 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 tolera
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 6≤ 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. Pöschel, 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 tolera
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 6≤ 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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
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. Pöschel, 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 tolera
x ∨y
K ∨ Lq
y
Lλ ∨ K
∇L
x
∇L
x ∧y
(1)
T (x, x ∨ y ) = T (x ∧ y , y )
Lλ
Lq
K = Quord(A, f )
∇L
L∗q
∇L
0L
Warsaw, June, 2014,
R. Pöschel, 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 tolera
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 .
1
L is not lower semimodular even for |A| = 3.
Warsaw, June, 2014,
R. Pöschel, 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 tolera
Warsaw, June, 2014,
R. Pöschel, 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 tolera
Warsaw, June, 2014,
R. Pöschel, The lattice of quasiorder lattices (20/20)