The Constraint Satisfaction Dichotomy Theorem for Beginners Tutorial – Part 2 Ross Willard University of Waterloo BLAST 2019 CU Boulder, May 22, 2019 Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 0 / 21
Recall: An algebra A = ( A , F ) is: idempotent if every f ∈ F satisfies ( ∀ x ) f ( x , x , . . . , x ) = x . Taylor if it is idempotent and has a term operation t ( x 1 , . . . , x n ) satisfying identities of the form ( ∀ x , y . . . ) t ( vars ) = t ( vars ′ ) forcing t to not be a projection. A (multi-sorted) CSP instance compatible with A consists of a family ( A x i : 1 ≤ i ≤ n ) of subalgebras of A (indexed by variables), and a set { C t : 1 ≤ t ≤ m } of “constraints” of the form R t ( x i 1 , . . . , x i k ) where R t ≤ sd A x i 1 × · · · × A x ik . Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 1 / 21
Assuming Θ is a CSP instance compatible with a Taylor algebra A and satisfying some level of local consistency, How can Θ nonetheless be inconsistent? One obvious way: if it encodes linear equations. Plan for today : to explain in detail how compatible subdirect relations of Taylor algebras encode linear equations. In particular, the role of: ◮ abelian congruences ◮ critical rectangular relations ◮ strands ◮ similarity Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 2 / 21
I will explain by examples, using “Maltsev reducts of groups.” Definition Given a group G , its Maltsev reduct is the algebra G aff = ( G , xy − 1 z ). Note: 1 G aff is Taylor. 2 G and G aff have the same congruences. 3 The relations compatible with G aff are any cosets (left or right) of subgroups H ≤ G × · · · × G . Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 3 / 21
Example 1: Z p We’ve already seen Z aff = ( Z p , x − y + z ). p Z p 1 (abelian) Con Z aff Norm Z p = so = p { 0 } 0 A relation compatible with Z aff is 2 L 111 = { ( x , y , z ) ∈ ( Z 2 ) 3 : x + y + z = 1 } . Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 4 / 21
Observe that the relation L 111 has the following properties: 1 L 111 is subdirect. 2 L 111 is “functional at every variable.” ◮ This is equivalent to L 111 being fork-free, where a fork is a pair of elements in the relation which disagree at exactly one coordinate. Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 5 / 21
Other properties of L 111 : 3 L 111 is indecomposable: there is no partition of its coordinates such that L 111 is the product of its projections onto the two subsets. 4 L 111 is maximal in the lattice of subuniverses of Z aff × Z aff × Z aff 2 . 2 2 The unique strand of this relation is { 0 , 1 } × { 0 , 1 } × { 0 , 1 } . Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 6 / 21
Example 2: S 3 Consider the symmetric group S 3 of order 6: � a , b | a 3 = b 2 = 1 , ab = ba − 1 � = S 3 { 1 , a , a 2 } ∪ { b , ba , ba 2 } . = S 3 1 Con S aff so ≡ N (abelian) Norm S 3 = = N 3 { 1 } 0 Let R ∗ = { ( x , y , z ) ∈ ( S 3 ) 3 : x ≡ N y ≡ N z } . For each c , d ∈ Z 3 let { ( a i , a j , a k ) : i + j + k = c (mod 3) } = R cd { ( ba i , ba j , ba k ) : i + j + k = d (mod 3) } . ∪ Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 7 / 21
Observe that: R 01 is subdirect, fork-free and indecomposable. R 01 supports two distinct (and disjoint) strands: N c × N c × N c . N × N × N and Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 8 / 21
{ ( a i , a j , a k ) : i + j + k = 0 (mod 3) } R 01 = { ( ba i , ba j , ba k ) : i + j + k = 1 (mod 3) } . ∪ One more property: R 01 is meet-irreducible in the subuniverse lattice of S aff × S aff × S aff 3 . 3 3 Proof sketch. Recall R ∗ = { ( x , y , z ) ∈ ( S 3 ) 3 : x ≡ N y ≡ N z } . Claim: R ∗ is the unique minimal subuniverse properly containing R 01 . First, it’s easy to see that R 01 is maximal in R ∗ . 3 ) 3 containing R 01 and some x �∈ R ∗ . Suppose B is a subuniverse of ( S aff WLOG, x = ( b , a , a 2 ). Also note that ( a , a , a ) ∈ R 01 . Then ( b , a , a 2 )( a , a , a ) − 1 ( b , a , a 2 ) = ( a , a , 1) ∈ B ∩ ( R ∗ \ R 01 ). Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 9 / 21
Using the R cd ’s, we can encode two systems of linear equations over Z 3 on parallel strands through cosets of N . From a CSP perspective, such parallel systems are easily solved. Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 10 / 21
Example 3: SL (2 , 5) Let G = SL (2 , 5) (the group of M ∈ Mat 2 × 2 ( Z 5 ) with det( M ) = 1). and G / Z ( G ) ∼ | G | = 120, Z ( G ) = { 1 , − 1 } , = A 5 . Let N = { 1 , − 1 } . SL (2 , 5) 1 Con G aff = so µ (abelian) Norm G = N { 1 } 0 Let G ( µ ) = { ( x , y ) ∈ G 2 : x µ y } ≤ G 2 . Define the map h : G ( µ ) → Z 2 by � 0 if x = y h (( x , y )) = 1 otherwise (i.e., x = − y ). It is a homomorphism G ( µ ) → Z 2 (because N is central). Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 11 / 21
Thus we can define G ( µ ) 3 R ∗ = { ( x , y , z ) ∈ G ( µ ) 3 : h ( x ) + h ( y ) + h ( z ) = 0 } R 0 = { ( x , y , z ) ∈ G ( µ ) 3 : h ( x ) + h ( y ) + h ( z ) = 1 } = R 1 all viewed as 6-ary relations compatible with G aff . Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 12 / 21
Properties of R 0 and R 1 : 1 Each is subdirect, fork-free and indecomposable. 2 Each is meet-irreducible in the subuniverse lattice of ( G aff ) 6 . R ∗ = G ( µ ) 3 is their common upper cover (exercise). 3 Each supports 3,600 distinct strands, each of the form A 2 × B 2 × C 2 where A , B , C are µ -classes (cosets of N ). 4 Restricted to any strand, R 0 or R 1 defines a linear equation. 5 The strands “cross” each other; CSPs do not parallelize this time. This is the interesting situation; doesn’t reduce to simpler scenarios. It turns out that strands being “fully linked” (like this example) is connected to the commutator condition [1 , µ ] = 0. Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 13 / 21
Summary of the 3 examples 1 L 111 ≤ Z aff × Z aff × Z aff 2 2 2 2 R 01 ≤ S aff × S aff × S aff 3 3 3 3 R 0 ≤ G aff × G aff × G aff × G aff × G aff × G aff where G = SL (2 , 5). Common properties: 1 1 Potatoes A are subdirectly irreducible (SI). Con A = µ 2 Relations R are compatible, subdirect. 0 3 Relations are fork-free. 4 Relations are indecomposable and meet-irreducible (= critical). 5 The minimal upper cover R ∗ of the relation R is the coordinatewise µ -closure of R ( µ = the monolith). 6 µ is “abelian.” Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 14 / 21
Centrality and the commutator Let A be any algebra. Let α, β ∈ Con A . There is a relation “ α centralizes β ” on congruences. [ α, β ] = 0 ⇐ ⇒ α centralizes β . α is “abelian” ⇐ ⇒ [ α, α ] = 0. For all β there is a largest α such that [ α, β ] = 0. This largest α is denoted (0 : β ) and called the annihilator of β . Examples: 1 Z aff p : monolith = 1, [1 , 1] = 0, (0 : 1) = 1. 2 S aff 3 : monolith = µ , [ µ, µ ] = 0, (0 : µ ) = µ . 3 SL (2 , 5) aff : monolith = µ , [ µ, µ ] = 0, (0 : µ ) = 1. Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 15 / 21
Theorem (comb. of Kearnes & Szendrei and Freese & McKenzie) Suppose A 1 , . . . , A n are finite algebras in an idempotent congruence modular variety with n ≥ 3. Assume R ≤ sd A 1 × · · · × A n and R is critical and fork-free, and let R ∗ be its unique upper cover. 1 Each A i is subdirectly irreducible with abelian monolith µ i . 2 R ∗ is the µ 1 × · · · × µ n -closure of R . 3 A i / (0 : µ i ) ∼ = A j / (0 : µ j ) for all i , j . 4 There exists a prime p such that each µ i -class (for any i ) has size a power of p . 5 If (0 : µ i ) = 1 for some (equivalently all) i , then: All µ i -classes (for all i ) have the same fixed size p k . 1 Each µ i -class can be identified with a k -dimensional vector space over 2 Z p , and with respect to these identifications, R restricted to any strand encodes k linear equations over Z p . Let A 1 ( µ 1 ) = µ 1 considered as a subalgebra of A 1 × A 1 . There exists a 3 simple affine algebra M with | M | = p k , and a surjective homomorphism A 1 ( µ 1 ) → M such that 0 A 1 is a kernel-class. Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 16 / 21
Almost the same thing can be proved in Taylor varieties. Theorem (TCT + last-minute help from Keith (thanks!)) Suppose A 1 , . . . , A n are finite algebras in an (idempotent) Taylor variety with n ≥ 3. Assume R ≤ sd A 1 × · · · × A n and R is critical and fork-free, and let R ∗ be its unique upper cover. 1 Each A i is subdirectly irreducible with abelian monolith µ i . 2 R ∗ is the µ 1 × · · · × µ n -closure of R . 3 A i / (0 : µ i ) ∼ = A j / (0 : µ j ) for all i , j . 4 There exists a prime p such that each µ i -class (for any i ) has size a power of p . 5 If (0 : µ i ) = 1 for some (equivalently all) i , then: All µ i -classes (for all i ) have the same fixed size p k . 1 Coordinatization? (Conjecture: something nice is true.) 2 There exists a simple affine algebra M with | M | = p m , and a surjective 3 homomorphism A 1 ( µ 1 ) → M , such that 0 A 1 is a kernel-class. Added May 24: see Lecture 3 for an improved statement. Ross Willard (Waterloo) CSP Dichotomy Theorem BLAST 2019 17 / 21
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