SLIDE 1
Finite generation and direct products
Peter Mayr & Nik Ruˇ skuc
JKU Linz, Austria University of St Andrews, UK
Warsaw, June 21, 2014
SLIDE 2 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).
SLIDE 3
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 A2 is not fg.
SLIDE 4
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 := (ZN, +, −, all constants) is generated by ∅, but A2 is not fg.
SLIDE 5 Proof, ⇐.
Let A = X, B = Y and x0 ∈ X, y0 ∈ Y . Z := X × {y0} ∪ {x0} × Y ∪ {(f A(x0, . . . , x0), y0) | | | f ∈ F} ∪ {(x0, f B(y0, . . . , y0)) | | | f ∈ F} Claim: ∀a ∈ A: (a, y0) ∈ Z Have term s over F and x1, . . . , xk ∈ X: sA(x1, . . . , xk) = a. Induct on length of s:
- 1. If s is a variable, then a = xi ∈ X and (a, y0) ∈ Z.
- 2. Assume s = f (t1, . . . , tn) for f ∈ F, terms t1, . . . , tn. For
ai := tA
i (x1, . . . , xk), we have (ai, y0) ∈ Z.
( f A(a1, . . . , an), f B(y0, . . . , y0) ) ∈ Z ( x0, f B(y0, . . . , y0) ) ∈ Z ( x0, y0 ) ∈ Z Applying the Mal’cev term in each row yields (a, y0) ∈ Z.
SLIDE 6
Proof, continued.
For all a ∈ A, b ∈ B (a, y0) ∈ Z (x0, y0) ∈ Z (x0, b) ∈ Z Applying the Mal’cev term in each row yields (a, b) ∈ Z. So A × B = Z.
SLIDE 7
Definition
A in a variety V is finitely presented (fp) if A ∼ = FV(x1, . . . , xk)/Cg ((r1, s1), . . . , (rn, sn)) for some k, n ∈ N and (r1, s1), . . . , (rn, sn) ∈ FV(x1, . . . , xk)2. In particular, fg free algebras are fp.
Theorem
In the variety V of loops with signature (·, \, /, 1): FV(x) × FV(x) is not fp.
Theorem
Let V be the variety of lattices, 2 := ({0, 1}, ∧, ∨). Then FV(x1, x2, x3) × 2 is not fp.
SLIDE 8 Proof, A ∈ V is not fp.
- 1. Find X finite and an epimorphism h: FV(X) → A.
- 2. Suppose ker h is generated by some (r1, s1), . . . , (rn, sn).
- 3. Find u, v ∈ FV(X) such that h(u) = h(v) in A but
u ≡ v in FV(X)/Cg ((r1, s1), . . . , (rn, sn)). 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).
SLIDE 9
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.
a1 ≡α a2 in A ⇒ (a1, b1) ≡α×1B (a2, b2) 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.
SLIDE 10
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 : d(x, y, y) ≡ x ≡ d(y, y, x) mod [Cg A(x, y), Cg A(x, y)] Each of the following implies a weak difference term: locally finite + Taylor, n-CP, CM
SLIDE 11
Proof, ⇒.
Let a1, a2 ∈ A be distinct, fix b ∈ B. Have ρ ∈ Con(A × B) of finite index and (a1, b) ≡ρ (a2, b). Show σ := {(u, v) ∈ A2 | | | ∃z ∈ B : (u, z) ≡ρ (v, z)}
◮ is a congruence on A, ◮ has finite index, and ◮ separates a1, a2
using commutators and the weak difference term.
SLIDE 12 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?
