Finite generation and direct products Peter Mayr & Nik Ru skuc - PowerPoint PPT Presentation

Finite generation and direct products Peter Mayr & Nik Ruˇ skuc JKU Linz, Austria University of St Andrews, UK Warsaw, June 21, 2014

Nice Boring Theorem A × B satisfies P ⇔ A and B satisfy P . Examples for property P : being finitely generated, finitely presented, residually finite ,... Example 1. Groups: G × H fg ⇔ G , H fg (same for fp, rf) 2. Semigroups: ( N , +) is fg, but ( N , +) 2 is not. Problem Which algebras and properties give Nice Boring Theorems? For semigroups: Robertson, Ruˇ skuc, Wiegold (1998), Gray, Ruˇ skuc (2009).

1. Finite generation Lemma (Folklore) A × B fg ⇒ A , B fg Proof. fg is inherited by homomorphic images. Theorem In any idempotent variety [ t ( x , . . . , x ) ≈ x for all terms t ]: A × B fg ⇔ A , B fg Proof. A = � X � , B = � Y � ⇒ A × B = � X × Y � Remark We have a NBT for lattices but not for their expansions: A := ( N , max , min , x + 1) is generated by 1, but A 2 is not fg.

Theorem (Geddes, PhD-thesis) In any Mal’cev variety of finite signature F : A × B fg ⇔ A , B fg Remark Finite signature is necessary. A := ( Z N , + , − , all constants) is generated by ∅ , but A 2 is not fg.

Proof, ⇐ . Let A = � X � , B = � Y � and x 0 ∈ X , y 0 ∈ Y . := X × { y 0 } ∪ { x 0 } × Y ∪ Z { ( f A ( x 0 , . . . , x 0 ) , y 0 ) | | | f ∈ F} ∪ { ( x 0 , f B ( y 0 , . . . , y 0 )) | | | f ∈ F} Claim: ∀ a ∈ A : ( a , y 0 ) ∈ � Z � Have term s over F and x 1 , . . . , x k ∈ X : s A ( x 1 , . . . , x k ) = a . Induct on length of s : 1. If s is a variable, then a = x i ∈ X and ( a , y 0 ) ∈ Z . 2. Assume s = f ( t 1 , . . . , t n ) for f ∈ F , terms t 1 , . . . , t n . For a i := t A i ( x 1 , . . . , x k ), we have ( a i , y 0 ) ∈ � Z � . f A ( a 1 , . . . , a n ) , f B ( y 0 , . . . , y 0 ) ( ) ∈ � Z � f B ( y 0 , . . . , y 0 ) ( x 0 , ) ∈ Z ( x 0 , ) ∈ Z y 0 Applying the Mal’cev term in each row yields ( a , y 0 ) ∈ � Z � .

Proof, continued. For all a ∈ A , b ∈ B ( a , y 0 ) ∈ � Z � ( x 0 , y 0 ) ∈ Z ( x 0 , b ) ∈ � Z � Applying the Mal’cev term in each row yields ( a , b ) ∈ � Z � . So A × B = � Z � .

2. Finite presentations Definition A in a variety V is finitely presented (fp) if A ∼ = F V ( x 1 , . . . , x k ) / Cg (( r 1 , s 1 ) , . . . , ( r n , s n )) for some k , n ∈ N and ( r 1 , s 1 ) , . . . , ( r n , s n ) ∈ F V ( x 1 , . . . , x k ) 2 . In particular, fg free algebras are fp. Theorem In the variety V of loops with signature ( · , \ , /, 1): F V ( x ) × F V ( x ) is not fp. Theorem Let V be the variety of lattices, 2 := ( { 0 , 1 } , ∧ , ∨ ). Then F V ( x 1 , x 2 , x 3 ) × 2 is not fp.

Proof, A ∈ V is not fp. 1. Find X finite and an epimorphism h : F V ( X ) → A . 2. Suppose ker h is generated by some ( r 1 , s 1 ) , . . . , ( r n , s n ). 3. Find u , v ∈ F V ( X ) such that h ( u ) = h ( v ) in A but u �≡ v in F V ( X ) / Cg (( r 1 , s 1 ) , . . . , ( r n , s n )) . Contradiction. For the word problem in 3. we use ◮ for loops: Evans’ confluent rewriting systems (1951). ◮ for lattices: Dean’s solution of the word problem (1964).

3. Residually finite Definition A is residually finite (rf) if ∀ a , b ∈ A , a � = b , ∃ ρ ∈ Con ( A ) : A /ρ is finite and a �≡ ρ b Lemma (Folklore) A , B rf ⇒ A × B rf Proof. a 1 �≡ α a 2 in A ⇒ ( a 1 , b 1 ) �≡ α × 1 B ( a 2 , b 2 ) in A × B The converse holds for example ◮ if A , B embed into A × B , NBT for algebras with idempotents (groups, monoids, lattices) ◮ if Con ( A × B ) = Con ( A ) × Con ( B ). NBT for congruence distributive varieties ( N , x + 1) × ( N , max( x − 1 , 0)) is rf, but ( N , max( x − 1 , 0)) is not.

Theorem In a variety with weak difference term: A × B rf ⇔ A , B rf Definition d is a weak difference term for V if ∀ A ∈ V , ∀ x , y ∈ A : mod [ Cg A ( x , y ) , Cg A ( x , y )] d ( x , y , y ) ≡ x ≡ d ( y , y , x ) Each of the following implies a weak difference term: locally finite + Taylor, n-CP, CM

Proof, ⇒ . Let a 1 , a 2 ∈ A be distinct, fix b ∈ B . Have ρ ∈ Con ( A × B ) of finite index and ( a 1 , b ) �≡ ρ ( a 2 , b ). Show σ := { ( u , v ) ∈ A 2 | | | ∃ z ∈ B : ( u , z ) ≡ ρ ( v , z ) } ◮ is a congruence on A , ◮ has finite index, and ◮ separates a 1 , a 2 using commutators and the weak difference term.

Problems 1. When is a subdirect product of fg lattices fg? 2. Does A × B fp ⇒ A , B fp? 3. Characterize the fp loops, lattices, . . . A , B such that A × B is fp. 4. Is the following decidable: Given fp semigroups A := � X | | | R � , B := � Y | | | S � . Is A × B fp? 5. Does A × B rf ⇒ A , B rf in varieties with Taylor term?