New algebraic insights from the solutions to the dichotomy - PowerPoint PPT Presentation

New algebraic insights from the solutions to the dichotomy conjecture What I learned from reading Dmitriy’s proof (of the CSP Dichotomy Theorem), Part 5 Ross Willard University of Waterloo Second Algebra Week Siena June 28, 2019 Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 0 / 44

Motivation R. Freese, “Subdirectly irreducible algebras in modular varieties,” Springer LNM 1004 (1982). Develops a notion of “similarity” between different subdirectly irreducible (SI) algebras with “abelian” monoliths (in CM varieties). D. Zhuk, ”A proof of CSP Dichotomy Conjecture,” arXiv:1704:01914 (2017) Develops a notion of “bridge” (between certain meet-irreducible congruences of possibly different algebras). Results which formally appear to be consequences of a (hypothetical) generalization of Freese’s theory to finite SIs in Taylor varieties. My goal: to find this generalization. Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 1 / 44

Plan I will: Carefully formulate and (partly) prove one such generalization. State some open problems. I will not: Explain Freese’s and Zhuk’s original results, or how my results generalize theirs. Mention CSP, polymorphisms, minions, etc. (promise!) I assume you: are comfortable with notions from universal algebra, . . . can tolerate 1.5 hours focused on an algebraic notion (“abelianness”) which never arises in the presence of lattice operations, and . . . are willing to accept ads for tame congruence theory (TCT). Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 2 / 44

Centrality (via the term condition) Definition. Let α, β, δ ∈ Con A . α centralizes β modulo δ ⇐ ⇒ ∀ term t ( x , y ), ∀ ( a i , b i ) ∈ α , ∀ ( c j , d j ) ∈ β , δ δ t ( a , c ) ≡ t ( a , d ) ⇐ ⇒ t ( b , c ) ≡ t ( b , d ) . β α � t ( a , c ) � t ( a , d ) Note: the 2 × 2 array is called an ( α, β )-matrix. t ( b , c ) t ( b , d ) More definitions: α centralizes β ⇐ ⇒ α centralizes β modulo 0. [ α, β ] = 0 ⇐ ⇒ α centralizes β . α is abelian ⇐ ⇒ [ α, α ] = 0. A is abelian ⇐ ⇒ [1 , 1] = 0. Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 3 / 44

Fact. Given any algebra A and any β ∈ Con A , there exists a unique largest α ∈ Con A which centralizes β . This α is called the centralizer of β and is denoted (0 : β ). Examples: 1 In the group ( Z 4 , +), if µ is the monolith, then (0 : µ ) = 1. Proof: ( Z 4 , +) is abelian, so [1 , 1] = 0, so [1 , µ ] = 0. 2 In the ring ( Z 4 , + , · ), with µ again the monolith, then (0 : µ ) = µ . Proof that (0 : µ ) � = 1: 1 · 0 � = 1 · 2 0 · 0 = 0 · 2 but µ 1 Thus 1 does not centralize µ . Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 4 / 44

Taylor varieties Definition. A variety V is Taylor if it satisfies either of the following equivalent conditions: 1 V satisfies some nontrivial idempotent Maltsev condition ( ≡ “satisfies a nontrivial set of idempotent identities” ` a la Julius). 2 V has a Taylor term, i.e., a term t ( x 1 , . . . , x n ) such that ◮ V | = t ( x , . . . , x ) ≈ x ( t is idempotent) ◮ For each i = 1 , . . . , n , V satisfies an identity of the form t ( vars , x , vars ′ ) ≈ t ( vars ′′ , y , vars ′′′ ) ↑ ↑ i i ( ≡ “satisfies a nontrivial idempotent loop condition” ` a la Julius). Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 5 / 44

Tame Congruence Theory If V is a locally finite Taylor variety, then: V “omits type 1.” V has a “weak near unanimity” (WNU) term. V has a “weak difference term.” Definition. A weak difference term is a term d ( x , y , z ) with the following property: For all A ∈ V and all α ∈ Con A , if α is abelian then d ( x , y , z ) “is Maltsev” on each α -class: ∀ ( a , b ) ∈ α, d ( a , a , b ) = b = d ( b , a , a ) . Intuition: d ( x , y , z ) | C = x − y + z for C ∈ A /α . Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 6 / 44

A classic construction Let A be a finite SI with monolith µ in a Taylor variety. Let µ be µ considered as a subalgebra of A 2 . Consider Con µ : 1 1 =1 θ θ Con µ = Con A = µ µ η 0 η 1 0 0 The kernels of the two projections: η 0 and η 1 For each θ ∈ Con A \ { 0 } , θ θ := { (( a 1 , a 2 ) , ( b 1 , b 2 )) ∈ µ × µ : a 1 ≡ b 1 } . Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 7 / 44

1 1=1 α α Con µ = Con A = ∆ µ µ η 0 η 1 0 0 Now assume that µ is abelian. Fix another congruence α such that α ≥ µ and [ α, µ ] = 0. Notation: For each α -class C , let 0 C := { ( a , a ) : a ∈ C } . Define ∆ = ∆ α,µ = Cg µ �� � �� � ( a , a ) , ( b , b ) : ( a , b ) ∈ α , i.e., the smallest congruence of µ collapsing each 0 C ( C ∈ A /α ). Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 8 / 44

∆ = Cg µ �� � �� � ( a , a ) , ( b , b ) : ( a , b ) ∈ α , i.e., collapsing 0 C ( C ∈ A /α ) 1 1=1 α α Con µ = Con A = ∆ µ µ η 0 η 1 ε 0 0 Let ε := ∆ ∧ µ . Easy facts: ∆ ≤ α . η 0 ∆ η 0 ∆ ∨ η 0 = α . Proof: ( a 1 , a 2 ) ≡ ( a 1 , a 1 ) ≡ ( b 1 , b 1 ) ≡ ( b 1 , b 2 ). α Similarly, ε ∨ η 0 = µ = ε ∨ η 1 . Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 9 / 44

1 1=1 α α Con µ = Con A = ∆ µ µ η 0 η 1 ε 0 0 ∆ ≡ ( a ′ , b ′ ) iff there exist Fact: in general, ( a , b ) ( a , b ) = ( a 0 , b 0 ) , ( a 1 , b 1 ) , . . . , ( a n , b n ) = ( a ′ , b ′ ) � a i − 1 � b i − 1 such that each is an ( α, µ )-matrix. a i b i Because [ α, µ ] = 0, we cannot have a i − 1 = b i − 1 and a i � = b i . Hence ∀ a ∈ A , ( a , a ) / ∆ ⊆ 0 C where C = a /α . I.e., 0 C is a ∆-class ( ∀ C ∈ A /α ). This proves ∆ < α and ε < µ . Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 10 / 44

Freese’s extra bit 1 1=1 α α Con µ = Con A = ∆ µ µ η 0 η 1 ε 0 0 Next goal: to show ∆ ≺ α . Aside: If Con µ were modular, this would be easy: ( η 0 , µ ) ց (0 , η 1 ). Hence 0 ≺ η 1 . Similarly, 0 ≺ η 0 so ε ≺ µ so ∆ ≺ α . Unfortunately, Con µ is probably not modular. Solution: computer-assisted proof! Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 11 / 44

1 Send email to Keith Kearnes: 2 Wait for answer: Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 12 / 44

1 1=1 α α Con µ = Con A = ∆ µ µ η 0 η 1 ε 0 0 Recall that A belongs to a Taylor variety V . Then by TCT: The covers ( η 0 , µ ) and ( η 1 , µ ) have type 2. So every cover between 0 and µ has type 2 (since 1 �∈ typ { V } ). So the interval I [0 , µ ] is modular. So ε ≺ µ as before. If ∆ �≺ α , then we would get an N 5 with abelian lower cover, impossible (as 1 �∈ typ { V } ). So ∆ ≺ α . Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 13 / 44

1 1=1 1 α =(0: µ ) α θ Con µ = Con A = ∆ Con D = µ µ 0 η 0 η 1 ε 0 0 Now assume that α = (0 : µ ). (the largest such that [ α, µ ] = 0) Then one can show ∆ is meet-irreducible and (∆ : α ) = α . Let D := µ / ∆ and θ = α/ ∆. D is SI, its monolith θ is abelian, (0 : θ ) = θ , and D /θ ∼ = A /α . Also, C ∈ A /α = ⇒ 0 C ∈ D . D o := { 0 C : C ∈ A /α } is a subuniverse of D . (Because 0 A ≤ µ .) D o is a transversal for θ . The natural map ν : µ → D satisfies ν − 1 ( D o ) = 0 A . Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 14 / 44

This proves most of: Theorem 1 Suppose A is a finite SI algebra with abelian monolith µ in a Taylor variety. Let α = (0 : µ ). There exists an SI algebra D with abelian monolith θ , a subuniverse D o ≤ D , a surjective homomorphism h : µ ։ D , and an isomorphism h ∗ : A /α ∼ = D /θ such that: 1 (0 : θ ) = θ . 2 D o is a transversal for θ . 3 h − 1 ( D o ) = 0 A . 4 h and h ∗ are compatible, i.e., h ( a , b ) /θ = h ∗ ( a /α ) = h ∗ ( b /α ). Moreover, ( D , D o ) is uniquely determined by A up to isomorphism. Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 15 / 44

Now forget how we constructed D (via Con µ , ∆). Focus on this: Given A finite SI in Taylor variety, abel. monolith µ , (0 : µ ) = α , (and h : µ ։ D and h ∗ : A /α ∼ ∃ essentially unique ( D , D o ) = D /θ ) s.t. . . . o 3 . h . . D (SI) µ o 2 . . h ∗ . o 1 ∼ = θ = monolith = (0 : θ ) D o = { o 1 , o 2 , o 3 } ≤ D A α classes Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 16 / 44

Example Let A = the quaternion group Q 8 = {± 1 , ± i , ± j , ± k } . Q 8 is SI, monolith µ is abelian. 1= α (0 : µ ) = 1. µ 0 µ has classes {± 1 } , {± i } , {± j } , {± k } . Theorem 1 is witnessed by the group D = ( Z 2 , +) and { o } = { 0 } : h : µ ։ D sends all ( x , x ) �→ 0 and all ( x , − x ) �→ 1. − k k − j h j 1 µ − i 0 i − 1 D 1 1 − 1 i − i j − j k − k Ross Willard (Waterloo) Algebraic insights What I learned Siena 2019 17 / 44