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 on a finite set Danica Jakub´ ıkov´ a-Studenovsk´ a Reinhard P¨ oschel 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¨ oschel, 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¨ oschel, 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¨ oschel, 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. ( a 1 , b 1 ) , . . . , ( a n , b n ) ∈ q = ⇒ ( f ( a 1 , . . . , a n ) , f ( b 1 , . . . , b n )) ∈ q . Pord � A , F � compatible partial orders (refl., trans., antisymmetric) Generalization of Pord � A , F � and Con � A , F � : Quord � A , F � compatible quasiorders (reflexive, transitive) Remark (Quord � A , 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 := ( { Quord � A , F � | F set of operations on A } , ⊆ ) . Warsaw, June, 2014, R. P¨ oschel, 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. ( a 1 , b 1 ) , . . . , ( a n , b n ) ∈ q = ⇒ ( f ( a 1 , . . . , a n ) , f ( b 1 , . . . , b n )) ∈ q . Pord � A , F � compatible partial orders (refl., trans., antisymmetric) Generalization of Pord � A , F � and Con � A , F � : Quord � A , F � compatible quasiorders (reflexive, transitive) Remark (Quord � A , 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 := ( { Quord � A , F � | F set of operations on A } , ⊆ ) . Warsaw, June, 2014, R. P¨ oschel, 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. ( a 1 , b 1 ) , . . . , ( a n , b n ) ∈ q = ⇒ ( f ( a 1 , . . . , a n ) , f ( b 1 , . . . , b n )) ∈ q . Pord � A , F � compatible partial orders (refl., trans., antisymmetric) Generalization of Pord � A , F � and Con � A , F � : Quord � A , F � compatible quasiorders (reflexive, transitive) Remark (Quord � A , 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 := ( { Quord � A , F � | F set of operations on A } , ⊆ ) . Warsaw, June, 2014, R. P¨ oschel, 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. ( a 1 , b 1 ) , . . . , ( a n , b n ) ∈ q = ⇒ ( f ( a 1 , . . . , a n ) , f ( b 1 , . . . , b n )) ∈ q . Pord � A , F � compatible partial orders (refl., trans., antisymmetric) Generalization of Pord � A , F � and Con � A , F � : Quord � A , F � compatible quasiorders (reflexive, transitive) Remark (Quord � A , 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 := ( { Quord � A , F � | F set of operations on A } , ⊆ ) . Warsaw, June, 2014, R. P¨ oschel, 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. ( a 1 , b 1 ) , . . . , ( a n , b n ) ∈ q = ⇒ ( f ( a 1 , . . . , a n ) , f ( b 1 , . . . , b n )) ∈ q . Pord � A , F � compatible partial orders (refl., trans., antisymmetric) Generalization of Pord � A , F � and Con � A , F � : Quord � A , F � compatible quasiorders (reflexive, transitive) Remark (Quord � A , 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 := ( { Quord � A , F � | F set of operations on A } , ⊆ ) . Warsaw, June, 2014, R. P¨ oschel, 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. ( a 1 , b 1 ) , . . . , ( a n , b n ) ∈ q = ⇒ ( f ( a 1 , . . . , a n ) , f ( b 1 , . . . , b n )) ∈ q . Pord � A , F � compatible partial orders (refl., trans., antisymmetric) Generalization of Pord � A , F � and Con � A , F � : Quord � A , F � compatible quasiorders (reflexive, transitive) Remark (Quord � A , 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 := ( { Quord � A , F � | F set of operations on A } , ⊆ ) . Warsaw, June, 2014, R. P¨ oschel, 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. ( a 1 , b 1 ) , . . . , ( a n , b n ) ∈ q = ⇒ ( f ( a 1 , . . . , a n ) , f ( b 1 , . . . , b n )) ∈ q . Pord � A , F � compatible partial orders (refl., trans., antisymmetric) Generalization of Pord � A , F � and Con � A , F � : Quord � A , F � compatible quasiorders (reflexive, transitive) Remark (Quord � A , 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 := ( { Quord � A , F � | F set of operations on A } , ⊆ ) . Warsaw, June, 2014, R. P¨ oschel, 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 )) Quord � A , F � = Quord � A , H � � Quord � A , H � = Quord � A , f � . f ∈ H Thus L = ( { Quord � A , H � | H ≤ A A } , ⊆ ). Description of L : look for ∧ - and ∨ -irreducible elements Remark: End − Quord is a Galois connection (induced by ⊲ ). Warsaw, June, 2014, R. P¨ oschel, 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 )) Quord � A , F � = Quord � A , H � � Quord � A , H � = Quord � A , f � . f ∈ H Thus L = ( { Quord � A , H � | H ≤ A A } , ⊆ ). Description of L : look for ∧ - and ∨ -irreducible elements Remark: End − Quord is a Galois connection (induced by ⊲ ). Warsaw, June, 2014, R. P¨ oschel, The lattice of quasiorder lattices (5/20)