IntroState based systemsFunctors and Coalgebras
Fuctor propertiesWeak Pullback PreservationFunctors parameterized by algebras
Free-algebra functorPreimage preservationWeak kernel preservation
Conclusion- and Breaking News
IntroState based systemsFunctors and Coalgebras
Functor propertiesWeak Pullback PreservationFunctors parameterized by algebras
Free-algebra functorPreimage preservationWeak kernel preservation
Conclusion- and Breaking News
Automataπ set of states,
Ξ£ input alphabet
β’ Acceptor:
β’ πΏ: π Γ Ξ£ β π
β’ π β π
+,-
1
0
0,1
1
+,-,0,10
+,-
π 3
π 4π 0 π 2
π 1
+,-
Ξ£ = {0,1, +, β}
+,-,0,1
e.g.-1011 accepted
+0110 not accepted
Automataπ set of states,
Ξ£ input alphabet
β’ Acceptor:
β’ πΏ: π Γ Ξ£ β π
β’ π β π
+,-
1
0
0,1
1
+,-,0,10
+,-
π 3
π 4π 0 π 2
π 1
π
πΞ£
+,-
Ξ£ = {0,1, +, β}
+,-,0,1
e.g.-1011 accepted
+0110 not accepted
Automataπ set of states,
Ξ£ input alphabet
β’ Acceptor:
β’ πΏ: π Γ Ξ£ β π
β’ π β π
+,-
1
0
0,1
1
+,-,0,10
+,-
π 3
π 4π 0 π 2
π 1
π π
πΞ£ 2
+,-
Ξ£ = {0,1, +, β}
+,-,0,1
e.g.-1011 accepted
+0110 not accepted
Automataπ set of states,
Ξ£ input alphabet
β’ Acceptor:
β’ πΏ: π Γ Ξ£ β π
β’ π β π
+,-
1
0
0,1
1
+,-,0,10
+,-
π 3
π 4π 0 π 2
π 1
π
πΞ£ Γ 2
+,-
Ξ£ = {0,1, +, β}
+,-,0,1
e.g.-1011 accepted
+0110 not accepted
Nondeterminism
β’ NFAβ’ πΏ: π Γ Ξ£ β β(π)
β’ π β π
a a
a, b
b b
ba
b
a
π 0
π 2
π 1 π 5π 3 π 4
Nondeterminism
β’ NFAβ’ πΏ: π Γ Ξ£ β β(π)
β’ π β π
a a
a, b
b b
ba
b
a
π
β(π)Ξ£
π 0
π 2
π 1 π 5π 3 π 4
Nondeterminism
β’ NFAβ’ πΏ: π Γ Ξ£ β β(π)
β’ π β π
a a
a, b
b b
ba
b
a
π
β(π)Ξ£ Γ 2
π 0
π 2
π 1 π 5π 3 π 4
Nondeterminism
β’ NFAβ’ πΏ: π Γ Ξ£ β β(π)
β’ π β π
β’ Kripke structureβ’ π β π Γ π
β’ π£ βΆ π β β(πΆ)
a a
a, b
b b
ba
b
a
π
β(π)Ξ£ Γ 2
π 0
π 2
π 1 π 5π 3 π 4
ππππππ’π
π π‘πππππ
π
β(π) Γ β(πΆ)
Nondeterminism
β’ NFAβ’ πΏ: π Γ Ξ£ β β(π)
β’ π β π
β’ Kripke structureβ’ π β π Γ π
β’ π£ βΆ π β β(πΆ)
a a
a, b
b b
ba
b
a
π
β(π)Ξ£ Γ 2
π 0
π 2
π 1 π 5π 3 π 4
ππππππ’π
π π‘πππππ
Probabilistic
β’ Probabilistic systems
β’ πΏ: π Γ π β 0,1 β
β’ Οπ₯βπ πΏ π , π₯ = 1
1/2
1/3
1/6
1/2 1/2
3/4
1/4
1/2
1/2
1π 0
π 1
π 2
π 3
π 4
π 5
Probabilistic
β’ Probabilistic systems
β’ πΏ: π Γ π β 0,1 β
β’ Οπ₯βπ πΏ π , π₯ = 1
1/2
1/3
1/6
1/2 1/2
3/4
1/4
1/2
1/2
1
π
π»(π)π 0
π 1
π 2
π 3
π 4
π 5
Higher order
β’ Neighbourhood structure
β’ π β π Γ 2π
β’ π: π β β(πΆ)
π
22πΓ β(πΆ)
Higher order
β’ Neighbourhood structure
β’ π β π Γ 2π
β’ π: π β β(πΆ)
β’ Topological space
β’ π β β β πβ’ closed under
β’ unionsβ’ finite intersections
π
22πΓ β(πΆ)
Higher order
β’ Neighbourhood structure
β’ π β π Γ 2π
β’ π: π β β(πΆ)
β’ Topological space
β’ π β β β πβ’ closed under
β’ unionsβ’ finite intersections
π
22πΓ β(πΆ)
π
π½ππ(π)
Coalgebras and Algebras
β’ Set functors fix the βsignatureβ of coalgebras
π
πΞ£ Γ 2
π
β(π)Ξ£ Γ 2
π
π»(π)
π
22πΓ β(πΆ)
π
π½ππ(π)
π
πΉ(π)
πΌ
Coalgebras and Algebras
β’ Set functors fix the βsignatureβ of coalgebras
β’ Algebras are upside-down coalgebras
π
πΉ(π)
πΌ
πΉ(π΄)
π΄
πΌπ΄ Γ π΄
π΄
π΄2 + π΄2
π΄
π΄2 + π΄ + 1
π΄
β(π΄)
π΄
β―
π
πΞ£ Γ 2
π
β(π)Ξ£ Γ 2
π
π»(π)
π
22πΓ β(πΆ)
π
π½ππ(π)
IntroState based systemsFunctors and Coalgebras
Fuctor propertiesWeak Pullback PreservationFunctors parameterized by algebras
Free-algebra functorPreimage preservationWeak kernel preservation
Conclusion- and Breaking News
Functors
β’ πΉ π a βnatural set theoretical constructionβ
β’ πΉ should be a functor:β’ for each π construct new set πΉ(π)
β’ for each π: π β π provide map πΉπ: πΉ π β πΉ(π)
such that
β’ πΉ π β π = πΉπ β πΉπ
β’ πΉ πππ΄ = πππΉ π΄
π
πΉ(π)
πΌ
Functors and Coalgebras
β’ πΉ π a βnatural set theoretical constructionβ
β’ πΉ should be a functor:β’ for each π construct new set πΉ(π)
β’ for each π: π β π provide map πΉπ: πΉ π β πΉ(π)
such that
β’ πΉ π β π = πΉπ β πΉπ
β’ πΉ πππ΄ = πππΉ π΄
π
πΉ(π)
πΌ
πΉ-coalgebra
πΉ-coalgebras and homomorphisms
π΅π΄
πΉ(π΄) πΉ(π΅)
πΌ π½
π
πΉπ
Homomorphism
πΉ-coalgebras and homomorphisms
π΅ πΆπ΄
πΉ(π΄) πΉ(π΅) πΉ(πΆ)
πΌ π½ πΎ
π
πΉπ
Homomorphisms compose
πΉ-coalgebras and homomorphisms
π΅ πΆπ΄
πΉ(π΄) πΉ(π΅) πΉ(πΆ)
πΌ π½ πΎ
π π
πΉπ πΉπ
Homomorphisms compose
πΉ-coalgebras and homomorphisms
π΅ πΆπ΄
πΉ(π΄) πΉ(π΅) πΉ(πΆ)
πΌ π½ πΎ
π π
π β π
πΉπ πΉπ
Homomorphisms compose
πΉ-coalgebras and homomorphisms
π΅ πΆπ΄
πΉ(π΄) πΉ(π΅) πΉ(πΆ)
πΌ π½ πΎ
π π
π β π
πΉ(π β π)
πΉπ πΉπ
Homomorphisms compose πΉ π β π = πΉπ β πΉπ
πππ‘πΉ
β’ πΉ-coalgebras form a category πππ‘πΉ
β’ Homomorphism theorems
β’ Substructures, quotients, congruences, β¦
β’ Co-equations, Co-Birkhoff
β’ Modal logic
β’ β¦
β’ Properties of πΉ determine structure of πππ‘πΉ- weak pullback preservation
π
πΉ(π)
IntroState based systemsFunctors and Coalgebras
Fuctor propertiesWeak Pullback PreservationFunctors parameterized by algebras
Free-algebra functorPreimage preservationWeak kernel preservation
Conclusion- and Breaking News
F weakly preserves pullbacks
β’ (Weak) pullback
A C
P B
f
g
π = { π, π β π΄ Γ π΅ β£ π π = π π }
F weakly preserves pullbacks
β’ (Weak) pullback
A C
P B
f
gπ1
π2π = { π, π β π΄ Γ π΅ β£ π π = π π }
F weakly preserves pullbacks
β’ (Weak) pullback
A C
P B
f
g
Q
π1
π2π π = { π, π β π΄ Γ π΅ β£ π π = π π }
F weakly preserves pullbacks
β’ (Weak) pullback
A C
P B
f
g
Q
π1
π2
π1
π1
π = { π, π β π΄ Γ π΅ β£ π π = π π }
F weakly preserves pullbacks
β’ (Weak) pullback
A C
P B
f
g
Q
π1
π2
π1
π1
π π = { π, π β π΄ Γ π΅ β£ π π = π π }
F weakly preserves pullbacks
β’ (Weak) pullback
β’ apply πΉ
A C
P B
F(A) F(C)
F(P) F(B)
f
g
Ff
Fg
π = { π, π β π΄ Γ π΅ β£ π π = π π }
F weakly preserves pullbacks
β’ (Weak) pullback
β’ apply πΉ
Is this a weak pullback diagram?
A C
P B
F(A) F(C)
F(P) F(B)
f
g
Ff
Fg
π = { π, π β π΄ Γ π΅ β£ π π = π π }
Observational equivalence
ππ΅ πΆ
πΉ(π΄)
πΉ(π΅)
π π΄
πΉ(πΆ)
πObservationalequivalence β¦
πΉ(π)
Observational equivalence
ππ΅ πΆ
πΉ(π΄)
πΉ(π΅)
π π΄
πΉ(πΆ)
πObservationalequivalence β¦
πΉ(π)
F weakly preserves pullbacks
ππ΅ πΆ
πΉ(π΄)
πΉ(π΅)
π π΄
πΉ(πΆ)
πObservationalequivalence β¦
πΉ(π)
Bisimilarity
ππ΅ πΆ
πΉ(π΄)
πΉ(π΅)
π π΄
πΉ(πΆ)
πObservationalequivalence β¦
β¦ is bisimilarity
Weak pullback preservation
β’ πΉ weakly preserves pullbacks, iff πΉ preservesβ’ kernel pairsβ’ and preimages
β’ πΉ weakly preserves kernel pairs, β’ iff congruences are bisimulationsβ’ iff observational equivalence = bisimilarity
π π
ker(π) π
π
π
Weak pullback preservation
β’ πΉ weakly preserves pullbacks, iff πΉ preservesβ’ kernel pairsβ’ and preimages
β’ πΉ weakly preserves kernel pairs, β’ iff congruences are bisimulationsβ’ iff observational equivalence = bisimilarity
β’ πΉ weakly preserves preimagesβ’ iff homomorphisms π: π΄ β π΅ + πΆ split domainβ’ iff π»π(π) = ππ»(π), for any class π
(provided |πΉ 1 | > 1 )
π π
ker(π) π
π
π
π π
πβ1 π π
π
π
Weak pullback preservation
β’ πΉ weakly preserves pullbacks, iff πΉ preservesβ’ kernel pairsβ’ and preimages
β’ πΉ weakly preserves kernel pairs, β’ iff πΉ preserves pullbacks of episβ’ iff observational equivalence = bisimilarity
π π
ker(π) π
π
π
Weak pullback preservation
β’ πΉ weakly preserves pullbacks, iff πΉ preservesβ’ kernel pairsβ’ and preimages
β’ πΉ weakly preserves kernel pairs, β’ iff πΉ preserves pullbacks of episβ’ iff observational equivalence = bisimilarity
β’ πΉ weakly preserves preimagesβ’ iff π»π(π) = ππ»(π), for any class π
(provided |πΉ 1 | > 1 )
π π
ker(π) π
π
π
π π
πβ1 π π
π
π
Functors weakly preserving special pullbacks
β’ all pullbacksβ’ πΉ π = β¦ π, ππ, Ξ£πβπΌπ
ππ , β π , βπ π , π½(π), π·π , β¦
Functors weakly preserving special pullbacks
β’ all pullbacksβ’ πΉ π = β¦ π, ππ, Ξ£πβπΌπ
ππ , β π , βπ π , π½(π), π·π , β¦
β’ preimagesβ’ πΉ π = β¦ β<π π , π2
3 , πΏπ , β¦
β’ kernel pairs
β’ πΉ π = β¦ 22π, π2 β π + 1, πππ(π),β¦
π π
πβ1 π π
π
π
Functors weakly preserving special pullbacks
β’ all pullbacksβ’ πΉ π = β¦ π, ππ, Ξ£πβπΌπ
ππ , β π , βπ π , π½(π), π·π , β¦
β’ preimagesβ’ πΉ π = β¦ β<π π , π2
3 , πΏπ , β¦
β’ kernel pairs
β’ πΉ π = β¦ 22π, π2 β π + 1, πππ π ,β¦ π π
ker(π) π
π
π
π π
πβ1 π π
π
π
IntroState based systemsFunctors and Coalgebras
Functor propertiesWeak Pullback PreservationFunctors parameterized by algebras
Free-algebra functorPreimage preservationWeak kernel preservation
Conclusion- and Breaking News
Parameterizing functors by algebras
β’ Let πΉπ depend on some algebra π
β’ Choose π so that πΉπ has desirable properties, e.g. (weakly) preserves
β’ kernel pairs
β’ preimages
β’ pullbacks
πΏ-fuzzy functor
1. Generalize β π = 2π
β’ Replace 2 = {0,1} by a complete lattice πΏ
β’ on objects: πΏπ
β’ on maps π: π β π:
πΏπ π (π¦):= α§
π π₯ =π¦
π(π₯)
For π β π put ππ π₯ = α1, π₯ β π0, π₯ β π
πΏ-fuzzy functor
1. Generalize β π = ππ
β’ Replace 2 = {0,1} by a complete lattice πΏ
β’ on objects: πΏπ
β’ on maps π: π β π:
πΏπ π (π¦):= α§
π π₯ =π¦
π(π₯)
β’ πΏ(β) always preserves preimages
For π β π put ππ π₯ = α1, π₯ β π0, π₯ β π
πΏ-fuzzy functor
1. Generalize β π = ππ
β’ Replace 2 = {0,1} by a complete lattice πΏ
β’ on objects: πΏπ
β’ on maps π: π β π:
πΏπ π (π¦):= α§
π π₯ =π¦
π(π₯)
β’ πΏ(β) always preserves preimages
β’ πΏ(β) weakly preserves kernel pairs iff πΏ is JID
π₯ β§ β π₯π π β πΌ = β{π₯ β§ π₯π β£ π β πΌ}
β―π₯π π₯π
π₯
For π β π put ππ π₯ = α1, π₯ β π0, π₯ β π
πΏ-fuzzy functor
1. Generalize β π = ππ
β’ Replace 2 = {0,1} by a complete lattice πΏ
β’ on objects: πΏπ
β’ on maps π: π β π:
πΏπ π (π¦):= α§
π π₯ =π¦
π(π₯)
β’ πΏ(β) always preserves preimages
β’ πΏ(β) weakly preserves kernel pairs iff πΏ is JID
π₯ β§ β π₯π π β πΌ = β{π₯ β§ π₯π β£ π β πΌ}
β―π₯π π₯π
π₯
For π β π put ππ π₯ = α1, π₯ β π0, π₯ β π
πΏ-fuzzy functor
1. Generalize β π = ππ
β’ Replace 2 = {0,1} by a complete lattice πΏ
β’ on objects: πΏπ
β’ on maps π: π β π:
πΏπ π (π¦):= α§
π π₯ =π¦
π(π₯)
β’ πΏ(β) always preserves preimages
β’ πΏ(β) weakly preserves kernel pairs iff πΏ is JID
π₯ β§ β π₯π π β πΌ = β{π₯ β§ π₯π β£ π β πΌ}
β―π₯π π₯π
π₯
β―
For π β π put ππ π₯ = α1, π₯ β π0, π₯ β π
πΏ-fuzzy functor
1. Generalize β π = ππ
β’ Replace 2 = {0,1} by a complete lattice πΏ
β’ on objects: πΏπ
β’ on maps π: π β π:
πΏπ π (π¦):= α§
π π₯ =π¦
π(π₯)
β’ πΏ(β) always preserves preimages
β’ πΏ(β) weakly preserves kernel pairs iff πΏ is JID
π₯ β§ β π₯π π β πΌ = β{π₯ β§ π₯π β£ π β πΌ}
β―π₯π π₯π
π₯
β―
For π β π put ππ π₯ = α1, π₯ β π0, π₯ β π
Finite π-bags
2. Generalize βπ(π)
β’ Replace boolean algebra π by a commutative monoidπ = (π,+, 0)
Finite π-bags
2. Generalize βπ(π)
β’ Replace boolean algebra π by a commutative monoidπ = (π,+, 0)
β’ on objects: πππ = {π1π₯1 +β―πππ₯π β£ π β β,ππ β π. π₯π β π}
Finite π-bags
2. Generalize βπ(π)
β’ Replace boolean algebra π by a commutative monoidπ = (π,+, 0)
β’ on objects: πππ = {π1π₯1 +β―πππ₯π β£ π β β,ππ β π. π₯π β π}
β’ on maps π: π β π: ππππ1π₯1 +β―πππ₯π βΆ= π1ππ₯1 +β―ππππ₯π
Finite π-bags
2. Generalize βπ(π)
β’ Replace boolean algebra π by a commutative monoidπ = (π,+, 0)
β’ on objects: πππ = {π1π₯1 +β―πππ₯π β£ π β β,ππ β π. π₯π β π}
β’ on maps π: π β π: ππππ1π₯1 +β―πππ₯π βΆ= π1ππ₯1 +β―ππππ₯π
β’ The functor ππ(β)
(weakly) preserves
Finite π-bags
2. Generalize βπ(π)
β’ Replace boolean algebra π by a commutative monoidπ = (π,+, 0)
β’ on objects: πππ = {π1π₯1 +β―πππ₯π β£ π β β,ππ β π. π₯π β π}
β’ on maps π: π β π: ππππ1π₯1 +β―πππ₯π βΆ= π1ππ₯1 +β―ππππ₯π
β’ The functor ππ(β)
(weakly) preserves
β’ preimages iff π is positive
Finite π-bags
2. Generalize βπ(π)
β’ Replace boolean algebra π by a commutative monoidπ = (π,+, 0)
β’ on objects: πππ = {π1π₯1 +β―πππ₯π β£ π β β,ππ β π. π₯π β π}
β’ on maps π: π β π: ππππ1π₯1 +β―πππ₯π βΆ= π1ππ₯1 +β―ππππ₯π
β’ The functor ππ(β)
(weakly) preserves
β’ preimages iff π is positive
positive: π₯ + π¦ = 0 β π₯ = 0
Finite π-bags
2. Generalize βπ(π)
β’ Replace boolean algebra π by a commutative monoidπ = (π,+, 0)
β’ on objects: πππ = {π1π₯1 +β―πππ₯π β£ π β β,ππ β π. π₯π β π}
β’ on maps π: π β π: ππππ1π₯1 +β―πππ₯π βΆ= π1ππ₯1 +β―ππππ₯π
β’ The functor ππ(β)
(weakly) preserves
β’ preimages iff π is positive
β’ kernel pairs iff π is refinable
positive: π₯ + π¦ = 0 β π₯ = 0
Finite π-bags
2. Generalize βπ(π)
β’ Replace boolean algebra π by a commutative monoidπ = (π,+, 0)
β’ on objects: πππ = {π1π₯1 +β―πππ₯π β£ π β β,ππ β π. π₯π β π}
β’ on maps π: π β π: ππππ1π₯1 +β―πππ₯π βΆ= π1ππ₯1 +β―ππππ₯π
β’ The functor ππ(β)
(weakly) preserves
β’ preimages iff π is positive
β’ kernel pairs iff π is refinable
refinable:π₯1 + π₯2 = π¦1 + π¦2 β
π₯1π₯2
π¦1 π¦2 =
positive: π₯ + π¦ = 0 β π₯ = 0
Finite π-bags
2. Generalize βπ(π)
β’ Replace boolean algebra π by a commutative monoidπ = (π,+, 0)
β’ on objects: πππ = {π1π₯1 +β―πππ₯π β£ π β β,ππ β π. π₯π β π}
β’ on maps π: π β π: ππππ1π₯1 +β―πππ₯π βΆ= π1ππ₯1 +β―ππππ₯π
β’ The functor ππ(β)
(weakly) preserves
β’ preimages iff π is positive
β’ kernel pairs iff π is refinable
positive: π₯ + π¦ = 0 β π₯ = 0
refinable:π₯1 + π₯2 = π¦1 + π¦2 β
π π π₯1π π π₯2π¦1 π¦2 =
IntroState based systemsFunctors and Coalgebras
Functor propertiesWeak Pullback PreservationFunctors parameterized by algebras
Free-algebra functorPreimage preservationWeak kernel preservation
Conclusion- and Breaking News
Properties of πΉΞ£(π)
πΉΞ£ π π
π
π
β ! ΰ΄€π
β’ Suppose π satisfies the equations Ξ£
β’ Each map π:π β π΄ has unique homomorphic extension
ΰ΄€π: πΉΞ£ π β π
Properties of πΉΞ£(π)
πΉΞ£ π π
π
π
β ! ΰ΄€π
β’ Suppose π satisfies the equations Ξ£
β’ Each map π:π β π΄ has unique homomorphic extension
ΰ΄€π: πΉΞ£ π β π
β’ ΰ΄€π π‘ π₯1, β¦ , π₯π = π‘π΄(ππ₯1, β¦ , ππ₯π)
Properties of πΉΞ£(π)
πΉΞ£ π πΉΞ£ π
π ππ
?
β’ Suppose π satisfies the equations Ξ£
β’ Each map π:π β π΄ has unique homomorphic extension
ΰ΄€π: πΉΞ£ π β π
β’ ΰ΄€π π‘ π₯1, β¦ , π₯π = π‘π΄(ππ₯1, β¦ , ππ₯π)
β’ πΉΞ£(β) is a πππ‘-functor
Properties of πΉΞ£(π)
πΉΞ£ π πΉΞ£ π
π π
ππ
π
?
β’ Suppose π satisfies the equations Ξ£
β’ Each map π:π β π΄ has unique homomorphic extension
ΰ΄€π: πΉΞ£ π β π
β’ ΰ΄€π π‘ π₯1, β¦ , π₯π = π‘π΄(ππ₯1, β¦ , ππ₯π)
β’ πΉΞ£(β) is a πππ‘-functor
Properties of πΉΞ£(π)
πΉΞ£ π πΉΞ£ π
π π
ππ
π
πβ²
β’ Suppose π satisfies the equations Ξ£
β’ Each map π:π β π΄ has unique homomorphic extension
ΰ΄€π: πΉΞ£ π β π
β’ ΰ΄€π π‘ π₯1, β¦ , π₯π = π‘π΄(ππ₯1, β¦ , ππ₯π)
β’ πΉΞ£(β) is a πππ‘-functor
Properties of πΉΞ£(π)
β’ Suppose π satisfies the equations Ξ£
β’ Each map π:π β π΄ has unique homomorphic extension
ΰ΄€π: πΉΞ£ π β π
β’ ΰ΄€π π‘ π₯1, β¦ , π₯π = π‘π΄(ππ₯1, β¦ , ππ₯π)
β’ πΉΞ£(β) is a πππ‘-functor
πΉΞ£ π πΉΞ£ π
π π
ππ
πΉΞ£π βΆ= ΰ΄₯πβ²
π
πβ²
IntroState based systemsFunctors and Coalgebras
Functor propertiesWeak Pullback PreservationFunctors parameterized by algebras
Free-algebra functorPreimage preservationWeak kernel preservation
Conclusion- and Breaking News
Independence
β’ π(π₯, π£1, β¦ , π£π) weakly independent of π₯ if
π π₯, π£1, β¦ , π£π β π(π¦) where π₯ β π¦
Independence
β’ π(π₯, π£1, β¦ , π£π) weakly independent of π₯ if
π π₯, π£1, β¦ , π£π β π(π¦) where π₯ β π¦
β’ π(π₯, π§1 , β¦ , π§π) independent of π₯, if
π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π) where π₯, π¦, π§1, β¦ , π§π mutually different
Independence
β’ π(π₯, π£1, β¦ , π£π) weakly independent of π₯ if
π π₯, π£1, β¦ , π£π β π(π¦) where π₯ β π¦
β’ π(π₯, π§1 , β¦ , π§π) independent of π₯, if
π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π) where π₯, π¦, π§1, β¦ , π§π mutually different
Derivative of Ξ£:
Ξ£β² β {π π₯, Τ¦π§ β π π¦, Τ¦π§ β£ π π₯, Τ¦π£ weakly independent of π₯}
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) π π₯, π£1, β¦ , π£π β π π¦ β’ π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) π π₯, π£1, β¦ , π£π β π π¦ β’ π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
π₯, π¦, π§1, β¦ , π§π π₯, π¦, π£1, β¦ , π£π
π₯ β πβ1 π¦ π¦
π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) π π₯, π£1, β¦ , π£π β π π¦ β’ π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
π₯, π¦, π§1, β¦ , π§π π₯, π¦, π£1, β¦ , π£π
π₯ β πβ1 π¦ π¦
π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) π π₯, π£1, β¦ , π£π β π π¦ β’ π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
π₯, π¦, π§1, β¦ , π§π π₯, π¦, π£1, β¦ , π£π
π₯ β πβ1 π¦ π¦
ΰ΄€ππ π₯, π§1, β¦ , π§π β π ππ₯, ππ§1, β¦ , ππ§π π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) π π₯, π£1, β¦ , π£π β π π¦ β’ π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
π₯, π¦, π§1, β¦ , π§π π₯, π¦, π£1, β¦ , π£π
π₯ β πβ1 π¦ π¦
ΰ΄€ππ π₯, π§1, β¦ , π§π β π ππ₯, ππ§1, β¦ , ππ§πβ π π₯, π£1, β¦ , π£π
π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) π π₯, π£1, β¦ , π£π β π π¦ β’ π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
π₯, π¦, π§1, β¦ , π§π π₯, π¦, π£1, β¦ , π£π
π₯ β πβ1 π¦ π¦
ΰ΄€ππ π₯, π§1, β¦ , π§π β π ππ₯, ππ§1, β¦ , ππ§πβ π π₯, π£1, β¦ , π£πβ π π¦
π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) π π₯, π£1, β¦ , π£π β π π¦ β’ π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
π₯, π¦, π§1, β¦ , π§π π₯, π¦, π£1, β¦ , π£π
π₯ β πβ1 π¦ π¦
ΰ΄€ππ π₯, π§1, β¦ , π§π β π ππ₯, ππ§1, β¦ , ππ§πβ π π₯, π£1, β¦ , π£πβ π π¦ β πΉΞ£ π¦
π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) π π₯, π£1, β¦ , π£π β π π¦ β’ π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
π₯, π¦, π§1, β¦ , π§π π₯, π¦, π£1, β¦ , π£π
π₯ β πβ1 π¦ π¦
ΰ΄€ππ π₯, π§1, β¦ , π§π β π ππ₯, ππ§1, β¦ , ππ§πβ π π₯, π£1, β¦ , π£πβ π π¦ β πΉΞ£ π¦
β π π₯, π§1, β¦ , π§π β ΰ΄€πβ1 (πΉΞ£ π¦ )
π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) π π₯, π£1, β¦ , π£π β π π¦ β’ π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
π₯, π¦, π§1, β¦ , π§π π₯, π¦, π£1, β¦ , π£π
π₯ β πβ1 π¦ π¦
ΰ΄€ππ π₯, π§1, β¦ , π§π β π ππ₯, ππ§1, β¦ , ππ§πβ π π₯, π£1, β¦ , π£πβ π π¦ β πΉΞ£ π¦
β π π₯, π§1, β¦ , π§π β ΰ΄€πβ1 (πΉΞ£ π¦ )= πΉΞ£(π
β1 π¦ )
π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) π π₯, π£1, β¦ , π£π β π π¦ β’ π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
π₯, π¦, π§1, β¦ , π§π π₯, π¦, π£1, β¦ , π£π
π₯ β πβ1 π¦ π¦
ΰ΄€ππ π₯, π§1, β¦ , π§π β π ππ₯, ππ§1, β¦ , ππ§πβ π π₯, π£1, β¦ , π£πβ π π¦ β πΉΞ£ π¦
β π π₯, π§1, β¦ , π§π β ΰ΄€πβ1 (πΉΞ£ π¦ )= πΉΞ£(π
β1 π¦ )β πΉΞ£( π¦, π§1, β¦ , π§π )
π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) π π₯, π£1, β¦ , π£π β π π¦ β’ π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
π₯, π¦, π§1, β¦ , π§π π₯, π¦, π£1, β¦ , π£π
π₯ β πβ1 π¦ π¦
ΰ΄€ππ π₯, π§1, β¦ , π§π β π ππ₯, ππ§1, β¦ , ππ§πβ π π₯, π£1, β¦ , π£πβ π π¦ β πΉΞ£ π¦
β π π₯, π§1, β¦ , π§π β ΰ΄€πβ1 (πΉΞ£ π¦ )= πΉΞ£(π
β1 π¦ )β πΉΞ£( π¦, π§1, β¦ , π§π )
β π π₯, π§1, β¦ , π§π β π π¦, π§1, β¦ , π§π
π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) π π₯, π£1, β¦ , π£π β π π¦ β’ π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
π₯, π¦, π§1, β¦ , π§π π₯, π¦, π£1, β¦ , π£π
π₯ β πβ1 π¦ π¦
ΰ΄€ππ π₯, π§1, β¦ , π§π β π ππ₯, ππ§1, β¦ , ππ§πβ π π₯, π£1, β¦ , π£πβ π π¦ β πΉΞ£ π¦
β π π₯, π§1, β¦ , π§π β ΰ΄€πβ1 (πΉΞ£ π¦ )= πΉΞ£(π
β1 π¦ )β πΉΞ£( π¦, π§1, β¦ , π§π )
β π π₯, π§1, β¦ , π§π β π π¦, π§1, β¦ , π§π β π π₯, π§1, β¦ , π§π
π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) π π₯, π£1, β¦ , π£π β π π¦ β’ π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
π₯, π¦, π§1, β¦ , π§π π₯, π¦, π£1, β¦ , π£π
π₯ β πβ1 π¦ π¦
ΰ΄€ππ π₯, π§1, β¦ , π§π β π ππ₯, ππ§1, β¦ , ππ§πβ π π₯, π£1, β¦ , π£πβ π π¦ β πΉΞ£ π¦
β π π₯, π§1, β¦ , π§π β ΰ΄€πβ1 (πΉΞ£ π¦ )= πΉΞ£(π
β1 π¦ )β πΉΞ£( π¦, π§1, β¦ , π§π )
β π π₯, π§1, β¦ , π§π β π π¦, π§1, β¦ , π§π β π π₯, π§1, β¦ , π§π β π(π¦, π§1, β¦ , π§π)
π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
π π
π π
ππ
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
π βͺ π {π₯, π¦}
π {π¦}
π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
πΉΞ£ π βͺ π πΉΞ£({π₯, π¦})
πΉΞ£(π) πΉΞ£({π¦})
ΰ΄€π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
πΉΞ£ π βͺ π πΉΞ£({π₯, π¦})
πΉΞ£(π) πΉΞ£({π¦})
Given π β πΉΞ£(π βͺ π) with ΰ΄€ππ β πΉΞ£ π¦ show π β πΉΞ£(π)
ΰ΄€π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
πΉΞ£ π βͺ π πΉΞ£({π₯, π¦})
πΉΞ£(π) πΉΞ£({π¦})
Given π β πΉΞ£(π βͺ π) with ΰ΄€ππ β πΉΞ£ π¦ show π β πΉΞ£(π)
ΰ΄€ππ = ΰ΄€ππ π₯1, β¦ , π₯π, π¦1, β¦ , π¦πΰ΄€π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
πΉΞ£ π βͺ π πΉΞ£({π₯, π¦})
πΉΞ£(π) πΉΞ£({π¦})
Given π β πΉΞ£(π βͺ π) with ΰ΄€ππ β πΉΞ£ π¦ show π β πΉΞ£(π)
ΰ΄€ππ = ΰ΄€ππ π₯1, β¦ , π₯π, π¦1, β¦ , π¦πΰ΄€π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
πΉΞ£ π βͺ π πΉΞ£({π₯, π¦})
πΉΞ£(π) πΉΞ£({π¦})
Given π β πΉΞ£(π βͺ π) with ΰ΄€ππ β πΉΞ£ π¦ show π β πΉΞ£(π)
ΰ΄€ππ = ΰ΄€ππ π₯1, β¦ , π₯π, π¦1, β¦ , π¦π= π π₯,β¦ , π₯, π¦, β¦ , π¦
ΰ΄€π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
πΉΞ£ π βͺ π πΉΞ£({π₯, π¦})
πΉΞ£(π) πΉΞ£({π¦})
Given π β πΉΞ£(π βͺ π) with ΰ΄€ππ β πΉΞ£ π¦ show π β πΉΞ£(π)
ΰ΄€ππ = ΰ΄€ππ π₯1, β¦ , π₯π, π¦1, β¦ , π¦π= π π₯,β¦ , π₯, π¦, β¦ , π¦ β π(π¦)
ΰ΄€π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
πΉΞ£ π βͺ π πΉΞ£({π₯, π¦})
πΉΞ£(π) πΉΞ£({π¦})
Given π β πΉΞ£(π βͺ π) with ΰ΄€ππ β πΉΞ£ π¦ show π β πΉΞ£(π)
ΰ΄€ππ π₯1, β¦ , π₯π, π¦1, β¦ , π¦π= π π₯,β¦ , π₯, π¦, β¦ , π¦ β π(π¦)
π π₯1, π₯2, β¦ , π₯π, π¦1, β¦ , π¦πββββ
ΰ΄€π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
πΉΞ£ π βͺ π πΉΞ£({π₯, π¦})
πΉΞ£(π) πΉΞ£({π¦})
Given π β πΉΞ£(π βͺ π) with ΰ΄€ππ β πΉΞ£ π¦ show π β πΉΞ£(π)
ΰ΄€ππ π₯1, β¦ , π₯π, π¦1, β¦ , π¦π= π π₯,β¦ , π₯, π¦, β¦ , π¦ β π(π¦)
π π₯1, π₯2, β¦ , π₯π, π¦1, β¦ , π¦πβ π π§1, π₯2, β¦ , π₯π, π¦1, β¦ , π¦πβββ
ΰ΄€π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
πΉΞ£ π βͺ π πΉΞ£({π₯, π¦})
πΉΞ£(π) πΉΞ£({π¦})
Given π β πΉΞ£(π βͺ π) with ΰ΄€ππ β πΉΞ£ π¦ show π β πΉΞ£(π)
ΰ΄€ππ π₯1, β¦ , π₯π, π¦1, β¦ , π¦π= π π₯,β¦ , π₯, π¦, β¦ , π¦ β π(π¦)
π π₯1, π₯2, β¦ , π₯π, π¦1, β¦ , π¦πβ π π§1, π₯2, β¦ , π₯π, π¦1, β¦ , π¦πβ π π§1, π§2, β¦ , π₯π, π¦1, β¦ , π¦πββ
ΰ΄€π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
πΉΞ£ π βͺ π πΉΞ£({π₯, π¦})
πΉΞ£(π) πΉΞ£({π¦})
Given π β πΉΞ£(π βͺ π) with ΰ΄€ππ β πΉΞ£ π¦ show π β πΉΞ£(π)
ΰ΄€ππ π₯1, β¦ , π₯π, π¦1, β¦ , π¦π= π π₯,β¦ , π₯, π¦, β¦ , π¦ β π(π¦)
π π₯1, π₯2, β¦ , π₯π, π¦1, β¦ , π¦πβ π π§1, π₯2, β¦ , π₯π, π¦1, β¦ , π¦πβ π π§1, π§2, β¦ , π₯π, π¦1, β¦ , π¦πβ β―β π π§1, π§2, β¦ , π§π, π¦1, β¦ , π¦π
ΰ΄€π
Preservation of preimages
β’ Thm.: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
πΉΞ£ π βͺ π πΉΞ£({π₯, π¦})
πΉΞ£(π) πΉΞ£({π¦})
Given π β πΉΞ£(π βͺ π) with ΰ΄€ππ β πΉΞ£ π¦ show π β πΉΞ£(π)
ΰ΄€ππ π₯1, β¦ , π₯π, π¦1, β¦ , π¦π= π π₯,β¦ , π₯, π¦, β¦ , π¦ β π(π¦)
π π₯1, π₯2, β¦ , π₯π, π¦1, β¦ , π¦πβ π π§1, π₯2, β¦ , π₯π, π¦1, β¦ , π¦πβ π π§1, π§2, β¦ , π₯π, π¦1, β¦ , π¦πβ β―β π π§1, π§2, β¦ , π§π, π¦1, β¦ , π¦π β π π¦, π¦, β¦ , π¦, π¦1, β¦ , π¦π
ΰ΄€π
Preservation of preimages
β’ Thm: πΉΞ£ β preserves preimages iff Ξ£ β’ Ξ£β²
β’ Proof(β) Enough to consider classifying preimages
πΉΞ£ π βͺ π πΉΞ£({π₯, π¦})
πΉΞ£(π) πΉΞ£({π¦})
Given π β πΉΞ£(π βͺ π) with ΰ΄€ππ β πΉΞ£ π¦ show π β πΉΞ£(π)
ΰ΄€ππ π₯1, β¦ , π₯π, π¦1, β¦ , π¦π= π π₯,β¦ , π₯, π¦, β¦ , π¦ β π(π¦)
π π₯1, π₯2, β¦ , π₯π, π¦1, β¦ , π¦πβ π π§1, π₯2, β¦ , π₯π, π¦1, β¦ , π¦πβ π π§1, π§2, β¦ , π₯π, π¦1, β¦ , π¦πβ β―β π π§1, π§2, β¦ , π§π, π¦1, β¦ , π¦π β π π¦, π¦, β¦ , π¦, π¦1, β¦ , π¦π β πΉΞ£(π)
ΰ΄€π
IntroState based systemsFunctors and Coalgebras
Functor propertiesWeak Pullback PreservationFunctors parameterized by algebras
Free-algebra functorPreimage preservationWeak kernel preservation
Conclusion- and Breaking News
Malβcev term
β’ Variety π±(Ξ£) = all algebras satisfying Ξ£
β’ Malβcev variety βΆβ βπ.
π₯ = π(π₯, π¦, π¦)
π π₯, π₯, π¦ = π¦
Π. Π. ΠΠ°Π»ΡΡΠ΅Π²1909-1967
Malβcev term
β’ Variety π±(Ξ£) = all algebras satisfying Ξ£
β’ Malβcev variety βΆβ βπ.
π₯ = π(π₯, π¦, π¦)
π π₯, π₯, π¦ = π¦
Π. Π. ΠΠ°Π»ΡΡΠ΅Π²1909-1967
Groups:π π₯, π¦, π§ = π₯ β π¦β1 β π§
Quasigroups: π π₯, π¦, π§ = (π₯/(π¦\y)) β (π¦\π§)
Rings: π π₯, π¦, π§ = π₯ β π¦ + π§
β¦
Malβcev term
β’ Variety π±(Ξ£) = all algebras satisfying Ξ£
β’ Malβcev variety βΆβ βπ.
π₯ = π(π₯, π¦, π¦)
π π₯, π₯, π¦ = π¦
β’ π βpermutable variety βΆβ βπ. βπ1, β¦ ,ππ .
π₯ = π1 π₯, π¦, π¦ ,ππ π₯, π₯, π¦ = ππ+1 π₯, π¦, π¦
ππ π₯, π₯, π¦ = π¦
Π. Π. ΠΠ°Π»ΡΡΠ΅Π²1909-1967
Groups:π π₯, π¦, π§ = π₯ β π¦β1 β π§
Quasigroups: π π₯, π¦, π§ = (π₯/(π¦\y)) β (π¦\π§)
Rings: π π₯, π¦, π§ = π₯ β π¦ + π§
β¦
Malβcev term
β’ Variety π±(Ξ£) = all algebras satisfying Ξ£
β’ Malβcev variety βΆβ βπ.
π₯ = π(π₯, π¦, π¦)
π π₯, π₯, π¦ = π¦
β’ π βpermutable variety βΆβ βπ. βπ1, β¦ ,ππ .
π₯ = π1 π₯, π¦, π¦ ,ππ π₯, π₯, π¦ = ππ+1 π₯, π¦, π¦
ππ π₯, π₯, π¦ = π¦
Π. Π. ΠΠ°Π»ΡΡΠ΅Π²1909-1967
Groups:π π₯, π¦, π§ = π₯ β π¦β1 β π§
Quasigroups: π π₯, π¦, π§ = (π₯/(π¦\y)) β (π¦\π§)
Rings: π π₯, π¦, π§ = π₯ β π¦ + π§
β¦
β¦ the above, and also β¦
β’ implication algebrasβ’ all congruence regular algebras
β¦
When does πΉΞ£(β) preserve kernel pairs
β’ Thm: If π± is Malβcev, then πΉΞ£ β weakly preserves kernel pairs
When does πΉΞ£(β) preserve kernel pairs
β’ Thm: If π± is Malβcev, then πΉΞ£ β weakly preserves kernel pairs
When does πΉΞ£(β) preserve kernel pairs
β’ Thm: If π± is Malβcev, then πΉΞ£ β weakly preserves kernel pairs
When does πΉΞ£(β) preserve kernel pairs
β’ Thm: If π± is Malβcev, then πΉΞ£ β weakly preserves kernel pairs
When does πΉΞ£(β) preserve kernel pairs
β’ Thm: If π± is Malβcev, then πΉΞ£ β weakly preserves kernel pairs
When does πΉΞ£(β) preserve kernel pairs
β’ Thm: If π± is Malβcev, then πΉΞ£ β weakly preserves kernel pairs
Malβcev β weak kernel preservationβ’ Thm: If π± is Malβcev, then πΉΞ£ β weakly preserves kernel pairs
Malβcev β weak kernel preservationβ’ Thm: If π± is Malβcev, then πΉΞ£ β weakly preserves kernel pairs
Malβcev β weak kernel preservationβ’ Thm: If π± is Malβcev, then πΉΞ£ β weakly preserves kernel pairs
Malβcev β weak kernel preservationβ’ Thm: If π± is Malβcev, then πΉΞ£ β weakly preserves kernel pairs
Malβcev β weak kernel preservationβ’ Thm: If π± is Malβcev, then πΉΞ£ β weakly preserves kernel pairs
Malβcev β weak kernel preservationβ’ Thm: If π± is Malβcev, then πΉΞ£ β weakly preserves kernel pairs
Malβcev β weak kernel preservationβ’ Thm: If π± is Malβcev, then πΉΞ£ β weakly preserves kernel pairs
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
π
ππ π
βΉ
π π, π, π, π π π, π, π
π π, π, π π π, π, π = π(π, π, π)π
π
π
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
π
ππ π
βΉ
π π, π, π, π π π, π, π
π π, π, π π π, π, π = π(π, π, π)π
π
π
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
π
ππ π
βΉ
π π, π, π, π π π, π, π
π π, π, π π π, π, π = π(π, π, π)π
ππ
π
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
π
ππ π
βΉ
π π, π, π, π π π, π, π
π π, π, π π π, π, π = π(π, π, π)π
ππ
π
π
π
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
β’ Thm: Malβcev β π βpermutable + πΉΞ£ weakly preserves kernel pairs
π₯ = π(π₯, π¦, π¦)π π₯, π₯, π¦ = π¦
π₯ = π1 π₯, π¦, π¦ ,ππ π₯, π₯, π¦ = ππ+1 π₯, π¦, π¦
ππ π₯, π₯, π¦ = π¦+WKPβ
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
β’ Thm: Malβcev β π βpermutable + πΉΞ£ weakly preserves kernel pairs
π₯ = π1 π₯, π¦, π¦ ,β―
ππβ1 π₯, π₯, π¦ = ππ π₯, π¦, π¦ = π(π₯, π¦, π¦)ππ π₯, π₯, π¦ = ππ+1 π₯, π¦, π¦
π π₯, π¦, π¦ = ππ+1 π₯, π₯, π¦ = ππ+2 π₯, π¦, π¦β―
ππ π₯, π₯, π¦ = π¦
Proof β :
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
β’ Thm: Malβcev β π βpermutable + πΉΞ£ weakly preserves kernel pairs
π₯ = π1 π₯, π¦, π¦ ,β―
ππβ1 π₯, π₯, π¦ = ππ π₯, π¦, π¦ = π(π₯, π¦, π¦)ππ π₯, π₯, π¦ = ππ+1 π₯, π¦, π¦
π π₯, π¦, π¦ = ππ+1 π₯, π₯, π¦ = ππ+2 π₯, π¦, π¦β―
ππ π₯, π₯, π¦ = π¦
Proof β :
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
β’ Thm: Malβcev β π βpermutable + πΉΞ£ weakly preserves kernel pairs
π₯ = π1 π₯, π¦, π¦ ,β―
ππβ1 π₯, π₯, π¦ = ππ π₯, π¦, π¦ = π(π₯, π¦, π¦)ππ π₯, π₯, π¦ = ππ+1 π₯, π¦, π¦
π π₯, π¦, π¦ = ππ+1 π₯, π₯, π¦ = ππ+2 π₯, π¦, π¦β―
ππ π₯, π₯, π¦ = π¦
Proof β :
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
β’ Thm: Malβcev β π βpermutable + πΉΞ£ weakly preserves kernel pairs
π₯ = π1 π₯, π¦, π¦ ,β―
ππβ1 π₯, π₯, π¦ = ππ π₯, π¦, π¦ = π(π₯, π¦, π¦)ππ π₯, π₯, π¦ = ππ+1 π₯, π¦, π¦
π π₯, π¦, π¦ = ππ+1 π₯, π₯, π¦ = ππ+2 π₯, π¦, π¦β―
ππ π₯, π₯, π¦ = π¦
Proof β :ππ π₯, π¦, π§ = π π₯, π¦, π§, π§ππ+1 π₯, π¦, π§ = π π₯, π₯, π¦, π§
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
β’ Thm: Malβcev β π βpermutable + πΉΞ£ weakly preserves kernel pairs
π₯ = π1 π₯, π¦, π¦ ,β―
ππβ1 π₯, π₯, π¦ = ππ π₯, π¦, π¦ = π(π₯, π¦, π¦)ππ π₯, π₯, π¦ = ππ+1 π₯, π¦, π¦
π π₯, π¦, π¦ = ππ+1 π₯, π₯, π¦ = ππ+2 π₯, π¦, π¦β―
ππ π₯, π₯, π¦ = π¦
Proof β :ππ π₯, π¦, π§ = π π₯, π¦, π§, π§ππ+1 π₯, π¦, π§ = π π₯, π₯, π¦, π§
π π₯, π¦, π§ β π (π₯, π¦, π¦, π§)
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
β’ Thm: Malβcev β π βpermutable + πΉΞ£ weakly preserves kernel pairs
π₯ = π1 π₯, π¦, π¦ ,β―
ππβ1 π₯, π₯, π¦ = ππ π₯, π¦, π¦ = π(π₯, π¦, π¦)ππ π₯, π₯, π¦ = ππ+1 π₯, π¦, π¦
π π₯, π¦, π¦ = ππ+1 π₯, π₯, π¦ = ππ+2 π₯, π¦, π¦β―
ππ π₯, π₯, π¦ = π¦
Proof β :ππ π₯, π¦, π§ = π π₯, π¦, π§, π§ππ+1 π₯, π¦, π§ = π π₯, π₯, π¦, π§
π π₯, π¦, π§ β π (π₯, π¦, π¦, π§)
π π₯, π¦, π¦ = π π₯, π¦, π¦, π¦= ππ(π₯, π¦, π¦)
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
β’ Thm: Malβcev β π βpermutable + πΉΞ£ weakly preserves kernel pairs
π₯ = π1 π₯, π¦, π¦ ,β―
ππβ1 π₯, π₯, π¦ = ππ π₯, π¦, π¦ = π(π₯, π¦, π¦)ππ π₯, π₯, π¦ = ππ+1 π₯, π¦, π¦
π π₯, π¦, π¦ = ππ+1 π₯, π₯, π¦ = ππ+2 π₯, π¦, π¦β―
ππ π₯, π₯, π¦ = π¦
Proof β :ππ π₯, π¦, π§ = π π₯, π¦, π§, π§ππ+1 π₯, π¦, π§ = π π₯, π₯, π¦, π§
π π₯, π¦, π§ β π (π₯, π¦, π¦, π§)
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
β’ Thm: Malβcev β π βpermutable + πΉΞ£ weakly preserves kernel pairs
π₯ = π1 π₯, π¦, π¦ ,β―
ππβ1 π₯, π₯, π¦ = ππ π₯, π¦, π¦ = π(π₯, π¦, π¦)ππ π₯, π₯, π¦ = ππ+1 π₯, π¦, π¦
π π₯, π¦, π¦ = ππ+1 π₯, π₯, π¦ = ππ+2 π₯, π¦, π¦β―
ππ π₯, π₯, π¦ = π¦
Proof β :ππ π₯, π¦, π§ = π π₯, π¦, π§, π§ππ+1 π₯, π¦, π§ = π π₯, π₯, π¦, π§
π π₯, π¦, π§ β π (π₯, π¦, π¦, π§)
π π₯, π₯, π¦ = π π₯, π₯, π₯, π¦= ππ+1(π₯, π₯, π¦)
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
β’ Thm: Malβcev β π βpermutable + πΉΞ£ weakly preserves kernel pairs
π₯ = π1 π₯, π¦, π¦ ,β―
ππβ1 π₯, π₯, π¦ = ππ π₯, π¦, π¦ = π(π₯, π¦, π¦)ππ π₯, π₯, π¦ = ππ+1 π₯, π¦, π¦
π π₯, π¦, π¦ = ππ+1 π₯, π₯, π¦ = ππ+2 π₯, π¦, π¦β―
ππ π₯, π₯, π¦ = π¦
Proof β :ππ π₯, π¦, π§ = π π₯, π¦, π§, π§ππ+1 π₯, π¦, π§ = π π₯, π₯, π¦, π§
π π₯, π¦, π§ β π (π₯, π¦, π¦, π§)
π π₯, π₯, π¦ = π π₯, π₯, π₯, π¦= ππ+1(π₯, π₯, π¦)
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
β’ Thm: Malβcev β π βpermutable + πΉΞ£ weakly preserves kernel pairs
π₯ = π1 π₯, π¦, π¦ ,β―
ππβ1 π₯, π₯, π¦ = ππ π₯, π¦, π¦ = π(π₯, π¦, π¦)ππ π₯, π₯, π¦ = ππ+1 π₯, π¦, π¦
π π₯, π¦, π¦ = ππ+1 π₯, π₯, π¦ = ππ+2 π₯, π¦, π¦β―
ππ π₯, π₯, π¦ = π¦
Proof β :ππ π₯, π¦, π§ = π π₯, π¦, π§, π§ππ+1 π₯, π¦, π§ = π π₯, π₯, π¦, π§
π π₯, π¦, π§ β π (π₯, π¦, π¦, π§)
π-permutable + wkp β Malβcev
β’ Lemma: If πΉΞ£ weakly preserves kernel pairs then for any terms π, π
π π₯, π₯, π¦ = π π₯, π¦, π¦ β β π .
π π₯, π¦, π§ = π π₯, π¦, π§, π§π π₯, π¦, π§ = π (π₯, π₯, π¦, π§)
β’ Thm: Malβcev β π βpermutable + πΉΞ£ weakly preserves kernel pairs
π₯ = π1 π₯, π¦, π¦ ,β―
ππβ1 π₯, π₯, π¦ = π(π₯, π¦, π¦)
π π₯, π₯, π¦ = ππ+2 π₯, π¦, π¦β―
ππ π₯, π₯, π¦ = π¦
Proof β :ππ π₯, π¦, π§ = π π₯, π¦, π§, π§ππ+1 π₯, π¦, π§ = π π₯, π₯, π¦, π§
π π₯, π¦, π§ β π (π₯, π¦, π¦, π§)
Conclusion
β’ πΉΞ£ preserves
β’ preimages β weak independence implies independence
β’ kernel pairs β π±(Ξ£) is Malβcevβ ( π-permutable β Malβcev )
Open: β ???
Distributive and modular varieties
β’ If πΉπ± π preserves kernel pairs then
β’ π± is congruence distributive iffβπ β β. βπ.π is πβary majority term, i.e.
π π₯,β¦ , π₯, π¦ = β― = π π₯,β¦ , π₯, π¦, π₯, β¦ , π₯ = β― = π π¦, π₯, β¦ , π₯ = π₯
β’ π± is congruence modular iffβπ β β. βπ. βπ.
π π₯,β¦ , π₯, π¦ = β― = π π₯,β¦ , π₯, π¦, π₯, β¦ , π₯ = β― = π π₯, π¦, π₯, β¦ , π₯ = π₯π π¦, π₯,β¦ , π₯ = π π₯, π₯, π¦π π₯, π¦, π¦ = π₯