Free-algebra functors as coalgebraic signatures

Free-algebra functors ascoalgebraic signatures

H. Peter Gumm

PALS Seminar Boulder 27.10.2020

IntroState based systemsFunctors and Coalgebras

Fuctor propertiesWeak Pullback PreservationFunctors parameterized by algebras

Free-algebra functorPreimage preservationWeak kernel preservation

Conclusion- and Breaking News


Functor propertiesWeak Pullback PreservationFunctors parameterized by algebras



Automata𝑆 set of states,

Σ input alphabet

• Acceptor:

• 𝛿: 𝑆 × Σ → 𝑆

• 𝑇 ⊆ 𝑆

+,-

1

0

0,1

1

+,-,0,10

+,-

𝑠3

𝑠4𝑠0 𝑠2

𝑠1

+,-

Σ = {0,1, +, −}

+,-,0,1

e.g.-1011 accepted

+0110 not accepted


Σ input alphabet

• Acceptor:

• 𝛿: 𝑆 × Σ → 𝑆

• 𝑇 ⊆ 𝑆

+,-

1

0

0,1

1

+,-,0,10

+,-

𝑠3

𝑠4𝑠0 𝑠2

𝑠1

𝑆

𝑆Σ

+,-

Σ = {0,1, +, −}

+,-,0,1

e.g.-1011 accepted

+0110 not accepted


Σ input alphabet

• Acceptor:

• 𝛿: 𝑆 × Σ → 𝑆

• 𝑇 ⊆ 𝑆

+,-

1

0

0,1

1

+,-,0,10

+,-

𝑠3

𝑠4𝑠0 𝑠2

𝑠1

𝑆 𝑆

𝑆Σ 2

+,-

Σ = {0,1, +, −}

+,-,0,1

e.g.-1011 accepted

+0110 not accepted


Σ input alphabet

• Acceptor:

• 𝛿: 𝑆 × Σ → 𝑆

• 𝑇 ⊆ 𝑆

+,-

1

0

0,1

1

+,-,0,10

+,-

𝑠3

𝑠4𝑠0 𝑠2

𝑠1

𝑆

𝑆Σ × 2

+,-

Σ = {0,1, +, −}

+,-,0,1

e.g.-1011 accepted

+0110 not accepted

Nondeterminism

• NFA• 𝛿: 𝑆 × Σ → ℙ(𝑆)

• 𝑇 ⊆ 𝑆

a a

a, b

b b

ba

b

a

𝑠0

𝑠2

𝑠1 𝑠5𝑠3 𝑠4

Nondeterminism

• NFA• 𝛿: 𝑆 × Σ → ℙ(𝑆)

• 𝑇 ⊆ 𝑆

a a

a, b

b b

ba

b

a

𝑆

ℙ(𝑆)Σ

𝑠0

𝑠2


Nondeterminism

• NFA• 𝛿: 𝑆 × Σ → ℙ(𝑆)

• 𝑇 ⊆ 𝑆

a a

a, b

b b

ba

b

a

𝑆

ℙ(𝑆)Σ × 2

𝑠0

𝑠2


Nondeterminism

• NFA• 𝛿: 𝑆 × Σ → ℙ(𝑆)

• 𝑇 ⊆ 𝑆

• Kripke structure• 𝑅 ⊆ 𝑆 × 𝑆

• 𝑣 ∶ 𝑆 → ℙ(𝐶)

a a

a, b

b b

ba

b

a

𝑆

ℙ(𝑆)Σ × 2

𝑠0

𝑠2


𝑟𝑒𝑑𝑏𝑙𝑢𝑒

𝑠𝑡𝑟𝑖𝑝𝑒𝑑

𝑆

ℙ(𝑆) × ℙ(𝐶)

Nondeterminism

• NFA• 𝛿: 𝑆 × Σ → ℙ(𝑆)

• 𝑇 ⊆ 𝑆

• Kripke structure• 𝑅 ⊆ 𝑆 × 𝑆

• 𝑣 ∶ 𝑆 → ℙ(𝐶)

a a

a, b

b b

ba

b

a

𝑆

ℙ(𝑆)Σ × 2

𝑠0

𝑠2


𝑟𝑒𝑑𝑏𝑙𝑢𝑒

𝑠𝑡𝑟𝑖𝑝𝑒𝑑

Probabilistic

• Probabilistic systems

• 𝛿: 𝑆 × 𝑆 → 0,1 ℝ

• σ𝑥∈𝑆 𝛿 𝑠, 𝑥 = 1

1/2

1/3

1/6

1/2 1/2

3/4

1/4

1/2

1/2

1𝑠0

𝑠1

𝑠2

𝑠3

𝑠4

𝑠5

Probabilistic

• Probabilistic systems

• 𝛿: 𝑆 × 𝑆 → 0,1 ℝ

• σ𝑥∈𝑆 𝛿 𝑠, 𝑥 = 1

1/2

1/3

1/6

1/2 1/2

3/4

1/4

1/2

1/2

1

𝑆

𝔻(𝑆)𝑠0

𝑠1

𝑠2

𝑠3

𝑠4

𝑠5

Higher order

• Neighbourhood structure

• 𝑅 ⊆ 𝑆 × 2𝑆

• 𝑐: 𝑆 → ℙ(𝐶)

Higher order


• 𝑅 ⊆ 𝑆 × 2𝑆

• 𝑐: 𝑆 → ℙ(𝐶)

𝑆

22𝑆× ℙ(𝐶)

Higher order


• 𝑅 ⊆ 𝑆 × 2𝑆

• 𝑐: 𝑆 → ℙ(𝐶)

• Topological space

• 𝜏 ⊆ ℙ ℙ 𝑆• closed under

• unions• finite intersections

𝑆

22𝑆× ℙ(𝐶)

Higher order


• 𝑅 ⊆ 𝑆 × 2𝑆

• 𝑐: 𝑆 → ℙ(𝐶)

• Topological space

• 𝜏 ⊆ ℙ ℙ 𝑆• closed under

• unions• finite intersections

𝑆

22𝑆× ℙ(𝐶)

𝑆

𝔽𝑖𝑙(𝑆)

Coalgebras and Algebras

• Set functors fix the „signature“ of coalgebras

𝑆

𝑆Σ × 2

𝑆

ℙ(𝑆)Σ × 2

𝑆

𝔻(𝑆)

𝑆

22𝑆× ℙ(𝐶)

𝑆

𝔽𝑖𝑙(𝑆)

𝑆

𝐹(𝑆)

𝛼

Coalgebras and Algebras

• Set functors fix the „signature“ of coalgebras

• Algebras are upside-down coalgebras

𝑆

𝐹(𝑆)

𝛼

𝐹(𝐴)

𝐴

𝛼𝐴 × 𝐴

𝐴

𝐴2 + 𝐴2

𝐴

𝐴2 + 𝐴 + 1

𝐴

ℙ(𝐴)

𝐴

⋯

𝑆

𝑆Σ × 2

𝑆

ℙ(𝑆)Σ × 2

𝑆

𝔻(𝑆)

𝑆

22𝑆× ℙ(𝐶)

𝑆

𝔽𝑖𝑙(𝑆)





Functors

• 𝐹 𝑋 a „natural set theoretical construction“

• 𝐹 should be a functor:• for each 𝑋 construct new set 𝐹(𝑋)

• for each 𝑓: 𝑋 → 𝑌 provide map 𝐹𝑓: 𝐹 𝑋 → 𝐹(𝑌)

such that

• 𝐹 𝑔 ∘ 𝑓 = 𝐹𝑔 ∘ 𝐹𝑓

• 𝐹 𝜄𝑑𝐴 = 𝜄𝑑𝐹 𝐴

𝑆

𝐹(𝑆)

𝛼

Functors and Coalgebras

• 𝐹 𝑋 a „natural set theoretical construction“

• 𝐹 should be a functor:• for each 𝑋 construct new set 𝐹(𝑋)

• for each 𝑓: 𝑋 → 𝑌 provide map 𝐹𝑓: 𝐹 𝑋 → 𝐹(𝑌)

such that

• 𝐹 𝑔 ∘ 𝑓 = 𝐹𝑔 ∘ 𝐹𝑓

• 𝐹 𝜄𝑑𝐴 = 𝜄𝑑𝐹 𝐴

𝑆

𝐹(𝑆)

𝛼

𝐹-coalgebra

𝐹-coalgebras

𝐴

𝐹(𝐴)

𝛼

Homomorphism

𝐹-coalgebras

𝐵𝐴

𝐹(𝐴) 𝐹(𝐵)

𝛼 𝛽

Homomorphism

𝐹-coalgebras and homomorphisms

𝐵𝐴


𝛼 𝛽

𝜑

Homomorphism


𝐵𝐴


𝛼 𝛽

𝜑

𝐹𝜑

Homomorphism


𝐵 𝐶𝐴

𝐹(𝐴) 𝐹(𝐵) 𝐹(𝐶)

𝛼 𝛽 𝛾

𝜑

𝐹𝜑

Homomorphisms compose


𝐵 𝐶𝐴


𝛼 𝛽 𝛾

𝜑 𝜓

𝐹𝜑 𝐹𝜓



𝐵 𝐶𝐴


𝛼 𝛽 𝛾

𝜑 𝜓

𝜓 ∘ 𝜑

𝐹𝜑 𝐹𝜓



𝐵 𝐶𝐴


𝛼 𝛽 𝛾

𝜑 𝜓

𝜓 ∘ 𝜑

𝐹(𝜓 ∘ 𝜑)

𝐹𝜑 𝐹𝜓

Homomorphisms compose 𝐹 𝜓 ∘ 𝜑 = 𝐹𝜓 ∘ 𝐹𝜑

𝑆𝑒𝑡𝐹

• 𝐹-coalgebras form a category 𝑆𝑒𝑡𝐹

• Homomorphism theorems

• Substructures, quotients, congruences, …

• Co-equations, Co-Birkhoff

• Modal logic

• …

• Properties of 𝐹 determine structure of 𝑆𝑒𝑡𝐹- weak pullback preservation

𝑆

𝐹(𝑆)





F weakly preserves pullbacks

• (Weak) pullback

A C

P B

f

g

𝑃 = { 𝑎, 𝑏 ∈ 𝐴 × 𝐵 ∣ 𝑓 𝑎 = 𝑔 𝑏 }


• (Weak) pullback

A C

P B

f

g𝜋1

𝜋2𝑃 = { 𝑎, 𝑏 ∈ 𝐴 × 𝐵 ∣ 𝑓 𝑎 = 𝑔 𝑏 }


• (Weak) pullback

A C

P B

f

g

Q

𝜋1

𝜋2𝑑 𝑃 = { 𝑎, 𝑏 ∈ 𝐴 × 𝐵 ∣ 𝑓 𝑎 = 𝑔 𝑏 }


• (Weak) pullback

A C

P B

f

g

Q

𝜋1

𝜋2

𝑞1

𝑞1

𝑃 = { 𝑎, 𝑏 ∈ 𝐴 × 𝐵 ∣ 𝑓 𝑎 = 𝑔 𝑏 }


• (Weak) pullback

A C

P B

f

g

Q

𝜋1

𝜋2

𝑞1

𝑞1

𝑑 𝑃 = { 𝑎, 𝑏 ∈ 𝐴 × 𝐵 ∣ 𝑓 𝑎 = 𝑔 𝑏 }


• (Weak) pullback

• apply 𝐹

A C

P B

F(A) F(C)

F(P) F(B)

f

g

Ff

Fg

𝑃 = { 𝑎, 𝑏 ∈ 𝐴 × 𝐵 ∣ 𝑓 𝑎 = 𝑔 𝑏 }


• (Weak) pullback

• apply 𝐹

Is this a weak pullback diagram?

A C

P B

F(A) F(C)

F(P) F(B)

f

g

Ff

Fg

𝑃 = { 𝑎, 𝑏 ∈ 𝐴 × 𝐵 ∣ 𝑓 𝑎 = 𝑔 𝑏 }

Observational equivalence

𝐵 𝐶

𝐹(𝐵) 𝐹(𝐶)

𝜓


𝜑𝐵 𝐶

𝐹(𝐴)

𝐹(𝐵)

𝐴

𝐹(𝐶)

𝜓


𝜑𝐵 𝐶

𝐹(𝐴)

𝐹(𝐵)

𝑃 𝐴

𝐹(𝐶)

𝜓Observationalequivalence …

𝐹(𝑃)


𝜑𝐵 𝐶

𝐹(𝐴)

𝐹(𝐵)

𝑃 𝐴

𝐹(𝐶)


𝐹(𝑃)


𝜑𝐵 𝐶

𝐹(𝐴)

𝐹(𝐵)

𝑃 𝐴

𝐹(𝐶)


𝐹(𝑃)

Bisimilarity

𝜑𝐵 𝐶

𝐹(𝐴)

𝐹(𝐵)

𝑃 𝐴

𝐹(𝐶)


… is bisimilarity

Weak pullback preservation

• 𝐹 weakly preserves pullbacks, iff 𝐹 preserves• kernel pairs• and preimages

• 𝐹 weakly preserves kernel pairs, • iff congruences are bisimulations• iff observational equivalence = bisimilarity

𝑋 𝑌

ker(𝑓) 𝑋

𝑓

𝑓



• 𝐹 weakly preserves kernel pairs, • iff congruences are bisimulations• iff observational equivalence = bisimilarity

• 𝐹 weakly preserves preimages• iff homomorphisms 𝜑: 𝐴 → 𝐵 + 𝐶 split domain• iff 𝐻𝑆(𝔎) = 𝑆𝐻(𝔎), for any class 𝔎

(provided |𝐹 1 | > 1 )

𝑋 𝑌

ker(𝑓) 𝑋

𝑓

𝑓

𝑋 𝑌

𝑓−1 𝑉 𝑉

𝑓

𝜄



• 𝐹 weakly preserves kernel pairs, • iff 𝐹 preserves pullbacks of epis• iff observational equivalence = bisimilarity

𝑋 𝑌

ker(𝑓) 𝑋

𝑓

𝑓



• 𝐹 weakly preserves kernel pairs, • iff 𝐹 preserves pullbacks of epis• iff observational equivalence = bisimilarity

• 𝐹 weakly preserves preimages• iff 𝐻𝑆(𝔎) = 𝑆𝐻(𝔎), for any class 𝔎

(provided |𝐹 1 | > 1 )

𝑋 𝑌

ker(𝑓) 𝑋

𝑓

𝑓

𝑋 𝑌

𝑓−1 𝑉 𝑉

𝑓

𝜄

Functors weakly preserving special pullbacks

• all pullbacks• 𝐹 𝑋 = … 𝑋, 𝑋𝑛, Σ𝑖∈𝐼𝑋

𝑛𝑖 , ℙ 𝑋 , ℙ𝜔 𝑋 , 𝔽(𝑋), 𝐷𝑋 , …



𝑛𝑖 , ℙ 𝑋 , ℙ𝜔 𝑋 , 𝔽(𝑋), 𝐷𝑋 , …

• preimages• 𝐹 𝑋 = … ℙ<𝑛 𝑋 , 𝑋2

3 , 𝐿𝑋 , …

• kernel pairs

• 𝐹 𝑋 = … 22𝑋, 𝑋2 − 𝑋 + 1, 𝑂𝑑𝑑(𝑋),…

𝑋 𝑌

𝑓−1 𝑉 𝑉

𝑓

𝜄



𝑛𝑖 , ℙ 𝑋 , ℙ𝜔 𝑋 , 𝔽(𝑋), 𝐷𝑋 , …

• preimages• 𝐹 𝑋 = … ℙ<𝑛 𝑋 , 𝑋2

3 , 𝐿𝑋 , …

• kernel pairs

• 𝐹 𝑋 = … 22𝑋, 𝑋2 − 𝑋 + 1, 𝑂𝑑𝑑 𝑋 ,… 𝑋 𝑌

ker(𝑓) 𝑋

𝑓

𝑓

𝑋 𝑌

𝑓−1 𝑉 𝑉

𝑓

𝜄





Parameterizing functors by algebras

• Let 𝐹𝒜 depend on some algebra 𝒜

• Choose 𝒜 so that 𝐹𝒜 has desirable properties, e.g. (weakly) preserves

• kernel pairs

• preimages

• pullbacks

𝐿-fuzzy functor

1. Generalize ℙ 𝑋 = 2𝑋

• Replace 2 = {0,1} by a complete lattice 𝐿

• on objects: 𝐿𝑋

• on maps 𝑓: 𝑋 → 𝑌:

𝐿𝑓 𝜎 (𝑦):= ሧ

𝑓 𝑥 =𝑦

𝜎(𝑥)

For 𝑈 ⊆ 𝑋 put 𝜒𝑈 𝑥 = ቊ1, 𝑥 ∈ 𝑈0, 𝑥 ∉ 𝑈

𝐿-fuzzy functor

1. Generalize ℙ 𝑋 = 𝟐𝑋





𝑓 𝑥 =𝑦

𝜎(𝑥)

• 𝐿(−) always preserves preimages


𝐿-fuzzy functor






𝑓 𝑥 =𝑦

𝜎(𝑥)


• 𝐿(−) weakly preserves kernel pairs iff 𝐿 is JID

𝑥 ∧ ⋁ 𝑥𝑖 𝑖 ∈ 𝐼 = ⋁{𝑥 ∧ 𝑥𝑖 ∣ 𝑖 ∈ 𝐼}

⋯𝑥𝑖 𝑥𝑗

𝑥


𝐿-fuzzy functor






𝑓 𝑥 =𝑦

𝜎(𝑥)



𝑥 ∧ ⋁ 𝑥𝑖 𝑖 ∈ 𝐼 = ⋁{𝑥 ∧ 𝑥𝑖 ∣ 𝑖 ∈ 𝐼}


𝑥


𝐿-fuzzy functor






𝑓 𝑥 =𝑦

𝜎(𝑥)



𝑥 ∧ ⋁ 𝑥𝑖 𝑖 ∈ 𝐼 = ⋁{𝑥 ∧ 𝑥𝑖 ∣ 𝑖 ∈ 𝐼}


𝑥

⋯


𝐿-fuzzy functor






𝑓 𝑥 =𝑦

𝜎(𝑥)



𝑥 ∧ ⋁ 𝑥𝑖 𝑖 ∈ 𝐼 = ⋁{𝑥 ∧ 𝑥𝑖 ∣ 𝑖 ∈ 𝐼}


𝑥

⋯


Finite 𝕄-bags

2. Generalize ℙ𝜔(𝑋)

• Replace boolean algebra 𝟐 by a commutative monoid𝕄 = (𝑀,+, 0)

Finite 𝕄-bags



• on objects: 𝕄𝜔𝑋 = {𝑚1𝑥1 +⋯𝑚𝑛𝑥𝑛 ∣ 𝑛 ∈ ℕ,𝑚𝑖 ∈ 𝑀. 𝑥𝑖 ∈ 𝑋}

Finite 𝕄-bags



• on objects: 𝕄𝜔𝑋 = {𝑚1𝑥1 +⋯𝑚𝑛𝑥𝑛 ∣ 𝑛 ∈ ℕ,𝑚𝑖 ∈ 𝕄. 𝑥𝑖 ∈ 𝑋}

• on maps 𝑓: 𝑋 → 𝑌: 𝕄𝜔𝑓𝑚1𝑥1 +⋯𝑚𝑛𝑥𝑛 ∶= 𝑚1𝑓𝑥1 +⋯𝑚𝑛𝑓𝑥𝑛

Finite 𝕄-bags





• The functor 𝕄𝜔(−)

(weakly) preserves

Finite 𝕄-bags






(weakly) preserves

• preimages iff 𝕄 is positive

Finite 𝕄-bags






(weakly) preserves


positive: 𝑥 + 𝑦 = 0 ⇒ 𝑥 = 0

Finite 𝕄-bags






(weakly) preserves


• kernel pairs iff 𝕄 is refinable


Finite 𝕄-bags






(weakly) preserves



refinable:𝑥1 + 𝑥2 = 𝑦1 + 𝑦2 ⇒

𝑥1𝑥2

𝑦1 𝑦2 =


Finite 𝕄-bags






(weakly) preserves




refinable:𝑥1 + 𝑥2 = 𝑦1 + 𝑦2 ⇒

𝑎 𝑏 𝑥1𝑐 𝑑 𝑥2𝑦1 𝑦2 =





Properties of 𝐹Σ(𝑋)

𝐹Σ 𝑋 𝒜

𝑋

𝜑

• Suppose 𝒜 satisfies the equations Σ


𝐹Σ 𝑋 𝒜

𝑋

𝜑

∃ ! ത𝜑


• Each map 𝜑:𝑋 → 𝐴 has unique homomorphic extension

ത𝜑: 𝐹Σ 𝑋 → 𝒜


𝐹Σ 𝑋 𝒜

𝑋

𝜑

∃ ! ത𝜑




• ത𝜑 𝑡 𝑥1, … , 𝑥𝑛 = 𝑡𝐴(𝜑𝑥1, … , 𝜑𝑥𝑛)


𝐹Σ 𝑋 𝐹Σ 𝑌

𝑋 𝑌𝜑

?




• ത𝜑 𝑡 𝑥1, … , 𝑥𝑛 = 𝑡𝐴(𝜑𝑥1, … , 𝜑𝑥𝑛)

• 𝐹Σ(−) is a 𝑆𝑒𝑡-functor



𝑋 𝑌

𝜄𝑌

𝜑

?




• ത𝜑 𝑡 𝑥1, … , 𝑥𝑛 = 𝑡𝐴(𝜑𝑥1, … , 𝜑𝑥𝑛)




𝑋 𝑌

𝜄𝑌

𝜑

𝜑′




• ത𝜑 𝑡 𝑥1, … , 𝑥𝑛 = 𝑡𝐴(𝜑𝑥1, … , 𝜑𝑥𝑛)






• ത𝜑 𝑡 𝑥1, … , 𝑥𝑛 = 𝑡𝐴(𝜑𝑥1, … , 𝜑𝑥𝑛)



𝑋 𝑌

𝜄𝑌

𝐹Σ𝜑 ∶= ഥ𝜑′

𝜑

𝜑′





Independence

• 𝑝(𝑥, 𝑣1, … , 𝑣𝑛) weakly independent of 𝑥 if

𝑝 𝑥, 𝑣1, … , 𝑣𝑛 ≈ 𝑞(𝑦) where 𝑥 ≠ 𝑦

Independence



• 𝑝(𝑥, 𝑧1 , … , 𝑧𝑛) independent of 𝑥, if

𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ≈ 𝑝(𝑦, 𝑧1, … , 𝑧𝑛) where 𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 mutually different

Independence



• 𝑝(𝑥, 𝑧1 , … , 𝑧𝑛) independent of 𝑥, if

𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ≈ 𝑝(𝑦, 𝑧1, … , 𝑧𝑛) where 𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 mutually different

Derivative of Σ:

Σ′ ≔ {𝑝 𝑥, Ԧ𝑧 ≈ 𝑝 𝑦, Ԧ𝑧 ∣ 𝑝 𝑥, Ԧ𝑣 weakly independent of 𝑥}

Preservation of preimages

• Thm: 𝐹Σ − preserves preimages iff Σ ⊢ Σ′



• Proof(⇒)



• Proof(⇒) 𝑝 𝑥, 𝑣1, … , 𝑣𝑛 ≈ 𝑞 𝑦 ⊢ 𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ≈ 𝑝(𝑦, 𝑧1, … , 𝑧𝑛)




𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 𝑥, 𝑦, 𝑣1, … , 𝑣𝑛

𝑥 ∉ 𝜑−1 𝑦 𝑦

𝜑




𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 𝑥, 𝑦, 𝑣1, … , 𝑣𝑛

𝑥 ∉ 𝜑−1 𝑦 𝑦

𝜑




𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 𝑥, 𝑦, 𝑣1, … , 𝑣𝑛

𝑥 ∉ 𝜑−1 𝑦 𝑦

ത𝜑𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ≈ 𝑝 𝜑𝑥, 𝜑𝑧1, … , 𝜑𝑧𝑛 𝜑




𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 𝑥, 𝑦, 𝑣1, … , 𝑣𝑛

𝑥 ∉ 𝜑−1 𝑦 𝑦

ത𝜑𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ≈ 𝑝 𝜑𝑥, 𝜑𝑧1, … , 𝜑𝑧𝑛≈ 𝑝 𝑥, 𝑣1, … , 𝑣𝑛

𝜑




𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 𝑥, 𝑦, 𝑣1, … , 𝑣𝑛

𝑥 ∉ 𝜑−1 𝑦 𝑦

ത𝜑𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ≈ 𝑝 𝜑𝑥, 𝜑𝑧1, … , 𝜑𝑧𝑛≈ 𝑝 𝑥, 𝑣1, … , 𝑣𝑛≈ 𝑞 𝑦

𝜑




𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 𝑥, 𝑦, 𝑣1, … , 𝑣𝑛

𝑥 ∉ 𝜑−1 𝑦 𝑦

ത𝜑𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ≈ 𝑝 𝜑𝑥, 𝜑𝑧1, … , 𝜑𝑧𝑛≈ 𝑝 𝑥, 𝑣1, … , 𝑣𝑛≈ 𝑞 𝑦 ∈ 𝐹Σ 𝑦

𝜑




𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 𝑥, 𝑦, 𝑣1, … , 𝑣𝑛

𝑥 ∉ 𝜑−1 𝑦 𝑦


⇒ 𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ∈ ത𝜑−1 (𝐹Σ 𝑦 )

𝜑




𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 𝑥, 𝑦, 𝑣1, … , 𝑣𝑛

𝑥 ∉ 𝜑−1 𝑦 𝑦


⇒ 𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ∈ ത𝜑−1 (𝐹Σ 𝑦 )= 𝐹Σ(𝜑

−1 𝑦 )

𝜑




𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 𝑥, 𝑦, 𝑣1, … , 𝑣𝑛

𝑥 ∉ 𝜑−1 𝑦 𝑦


⇒ 𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ∈ ത𝜑−1 (𝐹Σ 𝑦 )= 𝐹Σ(𝜑

−1 𝑦 )⊆ 𝐹Σ( 𝑦, 𝑧1, … , 𝑧𝑛 )

𝜑




𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 𝑥, 𝑦, 𝑣1, … , 𝑣𝑛

𝑥 ∉ 𝜑−1 𝑦 𝑦


⇒ 𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ∈ ത𝜑−1 (𝐹Σ 𝑦 )= 𝐹Σ(𝜑

−1 𝑦 )⊆ 𝐹Σ( 𝑦, 𝑧1, … , 𝑧𝑛 )

⇒ 𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ≈ 𝑟 𝑦, 𝑧1, … , 𝑧𝑛

𝜑




𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 𝑥, 𝑦, 𝑣1, … , 𝑣𝑛

𝑥 ∉ 𝜑−1 𝑦 𝑦


⇒ 𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ∈ ത𝜑−1 (𝐹Σ 𝑦 )= 𝐹Σ(𝜑

−1 𝑦 )⊆ 𝐹Σ( 𝑦, 𝑧1, … , 𝑧𝑛 )

⇒ 𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ≈ 𝑟 𝑦, 𝑧1, … , 𝑧𝑛 ≈ 𝑟 𝑥, 𝑧1, … , 𝑧𝑛

𝜑




𝑥, 𝑦, 𝑧1, … , 𝑧𝑛 𝑥, 𝑦, 𝑣1, … , 𝑣𝑛

𝑥 ∉ 𝜑−1 𝑦 𝑦


⇒ 𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ∈ ത𝜑−1 (𝐹Σ 𝑦 )= 𝐹Σ(𝜑

−1 𝑦 )⊆ 𝐹Σ( 𝑦, 𝑧1, … , 𝑧𝑛 )

⇒ 𝑝 𝑥, 𝑧1, … , 𝑧𝑛 ≈ 𝑟 𝑦, 𝑧1, … , 𝑧𝑛 ≈ 𝑟 𝑥, 𝑧1, … , 𝑧𝑛 ≈ 𝑝(𝑦, 𝑧1, … , 𝑧𝑛)

𝜑



• Proof(⇐)



• Proof(⇐) Enough to consider classifying preimages




𝑆 𝟐

𝑈 𝟏

𝜒𝑈




𝑋 ∪ 𝑌 {𝑥, 𝑦}

𝑌 {𝑦}

𝜑




𝐹Σ 𝑋 ∪ 𝑌 𝐹Σ({𝑥, 𝑦})

𝐹Σ(𝑌) 𝐹Σ({𝑦})

ത𝜑






Given 𝑝 ∈ 𝐹Σ(𝑋 ∪ 𝑌) with ത𝜑𝑝 ∈ 𝐹Σ 𝑦 show 𝑝 ∈ 𝐹Σ(𝑌)

ത𝜑







ത𝜑𝑝 = ത𝜑𝑝 𝑥1, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚ത𝜑







ത𝜑𝑝 = ത𝜑𝑝 𝑥1, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚ത𝜑







ത𝜑𝑝 = ത𝜑𝑝 𝑥1, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚= 𝑝 𝑥,… , 𝑥, 𝑦, … , 𝑦

ത𝜑







ത𝜑𝑝 = ത𝜑𝑝 𝑥1, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚= 𝑝 𝑥,… , 𝑥, 𝑦, … , 𝑦 ≈ 𝑞(𝑦)

ത𝜑







ത𝜑𝑝 𝑥1, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚= 𝑝 𝑥,… , 𝑥, 𝑦, … , 𝑦 ≈ 𝑞(𝑦)

𝑝 𝑥1, 𝑥2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈≈≈≈

ത𝜑








𝑝 𝑥1, 𝑥2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈ 𝑝 𝑧1, 𝑥2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈≈≈

ത𝜑








𝑝 𝑥1, 𝑥2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈ 𝑝 𝑧1, 𝑥2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈ 𝑝 𝑧1, 𝑧2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈≈

ത𝜑








𝑝 𝑥1, 𝑥2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈ 𝑝 𝑧1, 𝑥2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈ 𝑝 𝑧1, 𝑧2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈ ⋯≈ 𝑝 𝑧1, 𝑧2, … , 𝑧𝑛, 𝑦1, … , 𝑦𝑚

ത𝜑


• Thm.: 𝐹Σ − preserves preimages iff Σ ⊢ Σ′






𝑝 𝑥1, 𝑥2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈ 𝑝 𝑧1, 𝑥2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈ 𝑝 𝑧1, 𝑧2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈ ⋯≈ 𝑝 𝑧1, 𝑧2, … , 𝑧𝑛, 𝑦1, … , 𝑦𝑚 ≈ 𝑝 𝑦, 𝑦, … , 𝑦, 𝑦1, … , 𝑦𝑚

ത𝜑








𝑝 𝑥1, 𝑥2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈ 𝑝 𝑧1, 𝑥2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈ 𝑝 𝑧1, 𝑧2, … , 𝑥𝑛, 𝑦1, … , 𝑦𝑚≈ ⋯≈ 𝑝 𝑧1, 𝑧2, … , 𝑧𝑛, 𝑦1, … , 𝑦𝑚 ≈ 𝑝 𝑦, 𝑦, … , 𝑦, 𝑦1, … , 𝑦𝑚 ∈ 𝐹Σ(𝑌)

ത𝜑





Mal‘cev term

• Variety 𝒱(Σ) = all algebras satisfying Σ

Mal‘cev term


• Mal‘cev variety ∶⇔ ∃𝑚.

𝑥 = 𝑚(𝑥, 𝑦, 𝑦)

𝑚 𝑥, 𝑥, 𝑦 = 𝑦

А. И. Мальцев1909-1967

Mal‘cev term



𝑥 = 𝑚(𝑥, 𝑦, 𝑦)

𝑚 𝑥, 𝑥, 𝑦 = 𝑦


Groups:𝑚 𝑥, 𝑦, 𝑧 = 𝑥 ∗ 𝑦−1 ∗ 𝑧

Quasigroups: 𝑚 𝑥, 𝑦, 𝑧 = (𝑥/(𝑦\y)) ∗ (𝑦\𝑧)

Rings: 𝑚 𝑥, 𝑦, 𝑧 = 𝑥 − 𝑦 + 𝑧

…

Mal‘cev term



𝑥 = 𝑚(𝑥, 𝑦, 𝑦)

𝑚 𝑥, 𝑥, 𝑦 = 𝑦

• 𝑛 −permutable variety ∶⇔ ∃𝑘. ∃𝑚1, … ,𝑚𝑘 .

𝑥 = 𝑚1 𝑥, 𝑦, 𝑦 ,𝑚𝑖 𝑥, 𝑥, 𝑦 = 𝑚𝑖+1 𝑥, 𝑦, 𝑦

𝑚𝑘 𝑥, 𝑥, 𝑦 = 𝑦





…

Mal‘cev term



𝑥 = 𝑚(𝑥, 𝑦, 𝑦)

𝑚 𝑥, 𝑥, 𝑦 = 𝑦

• 𝑛 −permutable variety ∶⇔ ∃𝑘. ∃𝑚1, … ,𝑚𝑘 .







…

… the above, and also …

• implication algebras• all congruence regular algebras

…

When does 𝐹Σ(−) preserve kernel pairs

• Thm: If 𝒱 is Mal‘cev, then 𝐹Σ − weakly preserves kernel pairs











Mal‘cev ⇒ weak kernel preservation• Thm: If 𝒱 is Mal‘cev, then 𝐹Σ − weakly preserves kernel pairs







𝑛-permutable + wkp ⇔ Mal‘cev

• Lemma: If 𝐹Σ weakly preserves kernel pairs then for any terms 𝑝, 𝑞

𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.

𝑝 𝑥, 𝑦, 𝑧 = 𝑠 𝑥, 𝑦, 𝑧, 𝑧𝑞 𝑥, 𝑦, 𝑧 = 𝑠(𝑥, 𝑥, 𝑦, 𝑧)



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.


𝑑

𝑐𝑎 𝑏

⟹

𝑠 𝑎, 𝑏, 𝑐, 𝑑 𝑞 𝑏, 𝑐, 𝑑

𝑝 𝑎, 𝑏, 𝑐 𝑝 𝑏, 𝑏, 𝑐 = 𝑞(𝑏, 𝑐, 𝑐)𝜃

𝜓

𝜃



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.


𝑑

𝑐𝑎 𝑏

⟹


𝑝 𝑎, 𝑏, 𝑐 𝑝 𝑏, 𝑏, 𝑐 = 𝑞(𝑏, 𝑐, 𝑐)𝜃

𝜓

𝜃



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.


𝑑

𝑐𝑎 𝑏

⟹


𝑝 𝑎, 𝑏, 𝑐 𝑝 𝑏, 𝑏, 𝑐 = 𝑞(𝑏, 𝑐, 𝑐)𝜃

𝜓𝜓

𝜃



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.


𝑑

𝑐𝑎 𝑏

⟹


𝑝 𝑎, 𝑏, 𝑐 𝑝 𝑏, 𝑏, 𝑐 = 𝑞(𝑏, 𝑐, 𝑐)𝜃

𝜓𝜓

𝜃

𝜓

𝜃



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.


• Thm: Mal‘cev ⇔ 𝑛 −permutable + 𝐹Σ weakly preserves kernel pairs

𝑥 = 𝑚(𝑥, 𝑦, 𝑦)𝑚 𝑥, 𝑥, 𝑦 = 𝑦


𝑚𝑘 𝑥, 𝑥, 𝑦 = 𝑦+WKP⇔



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.



𝑥 = 𝑚1 𝑥, 𝑦, 𝑦 ,⋯

𝑚𝑖−1 𝑥, 𝑥, 𝑦 = 𝑚𝑖 𝑥, 𝑦, 𝑦 = 𝑚(𝑥, 𝑦, 𝑦)𝑚𝑖 𝑥, 𝑥, 𝑦 = 𝑚𝑖+1 𝑥, 𝑦, 𝑦

𝑚 𝑥, 𝑦, 𝑦 = 𝑚𝑖+1 𝑥, 𝑥, 𝑦 = 𝑚𝑖+2 𝑥, 𝑦, 𝑦⋯


Proof ⇐ :



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.



𝑥 = 𝑚1 𝑥, 𝑦, 𝑦 ,⋯


𝑚 𝑥, 𝑦, 𝑦 = 𝑚𝑖+1 𝑥, 𝑥, 𝑦 = 𝑚𝑖+2 𝑥, 𝑦, 𝑦⋯


Proof ⇐ :



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.



𝑥 = 𝑚1 𝑥, 𝑦, 𝑦 ,⋯


𝑚 𝑥, 𝑦, 𝑦 = 𝑚𝑖+1 𝑥, 𝑥, 𝑦 = 𝑚𝑖+2 𝑥, 𝑦, 𝑦⋯


Proof ⇐ :



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.



𝑥 = 𝑚1 𝑥, 𝑦, 𝑦 ,⋯


𝑚 𝑥, 𝑦, 𝑦 = 𝑚𝑖+1 𝑥, 𝑥, 𝑦 = 𝑚𝑖+2 𝑥, 𝑦, 𝑦⋯


Proof ⇐ :𝑚𝑖 𝑥, 𝑦, 𝑧 = 𝑠 𝑥, 𝑦, 𝑧, 𝑧𝑚𝑖+1 𝑥, 𝑦, 𝑧 = 𝑠 𝑥, 𝑥, 𝑦, 𝑧



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.



𝑥 = 𝑚1 𝑥, 𝑦, 𝑦 ,⋯


𝑚 𝑥, 𝑦, 𝑦 = 𝑚𝑖+1 𝑥, 𝑥, 𝑦 = 𝑚𝑖+2 𝑥, 𝑦, 𝑦⋯



𝑚 𝑥, 𝑦, 𝑧 ≔ 𝑠(𝑥, 𝑦, 𝑦, 𝑧)



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.



𝑥 = 𝑚1 𝑥, 𝑦, 𝑦 ,⋯


𝑚 𝑥, 𝑦, 𝑦 = 𝑚𝑖+1 𝑥, 𝑥, 𝑦 = 𝑚𝑖+2 𝑥, 𝑦, 𝑦⋯



𝑚 𝑥, 𝑦, 𝑧 ≔ 𝑠(𝑥, 𝑦, 𝑦, 𝑧)

𝑚 𝑥, 𝑦, 𝑦 = 𝑠 𝑥, 𝑦, 𝑦, 𝑦= 𝑚𝑖(𝑥, 𝑦, 𝑦)



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.



𝑥 = 𝑚1 𝑥, 𝑦, 𝑦 ,⋯


𝑚 𝑥, 𝑦, 𝑦 = 𝑚𝑖+1 𝑥, 𝑥, 𝑦 = 𝑚𝑖+2 𝑥, 𝑦, 𝑦⋯



𝑚 𝑥, 𝑦, 𝑧 ≔ 𝑠(𝑥, 𝑦, 𝑦, 𝑧)



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.



𝑥 = 𝑚1 𝑥, 𝑦, 𝑦 ,⋯


𝑚 𝑥, 𝑦, 𝑦 = 𝑚𝑖+1 𝑥, 𝑥, 𝑦 = 𝑚𝑖+2 𝑥, 𝑦, 𝑦⋯



𝑚 𝑥, 𝑦, 𝑧 ≔ 𝑠(𝑥, 𝑦, 𝑦, 𝑧)

𝑚 𝑥, 𝑥, 𝑦 = 𝑠 𝑥, 𝑥, 𝑥, 𝑦= 𝑚𝑖+1(𝑥, 𝑥, 𝑦)



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.



𝑥 = 𝑚1 𝑥, 𝑦, 𝑦 ,⋯


𝑚 𝑥, 𝑦, 𝑦 = 𝑚𝑖+1 𝑥, 𝑥, 𝑦 = 𝑚𝑖+2 𝑥, 𝑦, 𝑦⋯



𝑚 𝑥, 𝑦, 𝑧 ≔ 𝑠(𝑥, 𝑦, 𝑦, 𝑧)

𝑚 𝑥, 𝑥, 𝑦 = 𝑠 𝑥, 𝑥, 𝑥, 𝑦= 𝑚𝑖+1(𝑥, 𝑥, 𝑦)



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.



𝑥 = 𝑚1 𝑥, 𝑦, 𝑦 ,⋯


𝑚 𝑥, 𝑦, 𝑦 = 𝑚𝑖+1 𝑥, 𝑥, 𝑦 = 𝑚𝑖+2 𝑥, 𝑦, 𝑦⋯



𝑚 𝑥, 𝑦, 𝑧 ≔ 𝑠(𝑥, 𝑦, 𝑦, 𝑧)



𝑝 𝑥, 𝑥, 𝑦 = 𝑞 𝑥, 𝑦, 𝑦 ⇔ ∃ 𝑠.



𝑥 = 𝑚1 𝑥, 𝑦, 𝑦 ,⋯

𝑚𝑖−1 𝑥, 𝑥, 𝑦 = 𝑚(𝑥, 𝑦, 𝑦)

𝑚 𝑥, 𝑥, 𝑦 = 𝑚𝑖+2 𝑥, 𝑦, 𝑦⋯



𝑚 𝑥, 𝑦, 𝑧 ≔ 𝑠(𝑥, 𝑦, 𝑦, 𝑧)

Conclusion

• 𝐹Σ preserves

• preimages ⇔ weak independence implies independence

• kernel pairs ⇐ 𝒱(Σ) is Mal‘cev⇒ ( 𝑛-permutable ⇒ Mal‘cev )

Open: ⇔ ???

Distributive and modular varieties

• If 𝐹𝒱 𝑋 preserves kernel pairs then

• 𝒱 is congruence distributive iff∃𝑘 ∈ ℕ. ∃𝑚.𝑚 is 𝑘−ary majority term, i.e.

𝑚 𝑥,… , 𝑥, 𝑦 = ⋯ = 𝑚 𝑥,… , 𝑥, 𝑦, 𝑥, … , 𝑥 = ⋯ = 𝑚 𝑦, 𝑥, … , 𝑥 = 𝑥

• 𝒱 is congruence modular iff∃𝑘 ∈ ℕ. ∃𝑚. ∃𝑞.

𝑚 𝑥,… , 𝑥, 𝑦 = ⋯ = 𝑚 𝑥,… , 𝑥, 𝑦, 𝑥, … , 𝑥 = ⋯ = 𝑚 𝑥, 𝑦, 𝑥, … , 𝑥 = 𝑥𝑚 𝑦, 𝑥,… , 𝑥 = 𝑞 𝑥, 𝑥, 𝑦𝑞 𝑥, 𝑦, 𝑦 = 𝑥

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Free-algebra functors as coalgebraic signatures