SLIDE 1 Lattices allowing only nilpotent commutator
Erhard Aichinger
Institute for Algebra Johannes Kepler University Linz, Austria
February 2017, AAA93
Supported by the Austrian Science Fund (FWF) : P24077 and P29931
SLIDE 2
Does L force nilpotency?
Question
◮ Given: A modular lattice L. ◮ Asked: Is there an algebra A
in a congruence modular variety with Con(A) ∼ = L such that A is not nilpotent? 1
SLIDE 3
Towards a purely lattice theoretic viewpoint
What a non-nilpotent algebra does to a finite lattice
If there is a non-nilpotent A in a cm variety with Con(A) = L, then the binary commutator operation of A [., .] : L × L → L satisfies
◮ ∀x, y : [x, y] = [y, x] ≤ x ∧ y, ◮ ∀x, y, z : [x, y ∨ z] = [x, y] ∨ [x, z]
and there is a nilpotency killer ρ ∈ L with
◮ ρ > 0. ◮ [1, ρ] = ρ.
SLIDE 4
Lattice theoretic question
An obvious dichotomy
Given a lattice L,
◮ there exists a commutative, join distributive,
“subintersective” binary operation [., .] that has a ρ ∈ L with [1, ρ] = ρ > 0, or
◮ there is no such operation.
Definition
A finite lattice L forces nilpotent type if there are no [., .] and ρ such that
◮ [., .] is commutative, join distributive, subintersective (i.e.,
[., .] is a commutator multiplication), and
◮ [1, ρ] = ρ > 0.
SLIDE 5
Lattice theoretic question
Goal
Characterize those finite modular lattices that force nilpotent type.
Very short history
◮ G. Birkhoff (1948) defined commutation lattices (L, ∨, ∧, (xy)).
Proved: if lower central series is finite, then the upper central series has the same length.
◮ J. Czelakowski (2008) defined commutator lattices
(L, ∨, ∧, [x, y]) and investigated the relation of [x, y] with (a : b) = largest c with [c, b] ≤ a.
◮ At AAA92 (2016), we saw a condition (C) such that every finite
modular lattice with (C) forces nilpotent type.
◮ Today, we prove the converse and thereby finish the
characterization for finite modular lattices.
SLIDE 6
Construction of a commutator multiplication
Task
◮ Given: L. ◮ Asked: A multiplication [., .] and
a nilpotency killer ρ.
Finding [., .] and ρ
1
SLIDE 7
Construction of a commutator multiplication
Task
◮ Given: L. ◮ Asked: A multiplication [., .] and
a nilpotency killer ρ.
Finding [., .] and ρ
◮ We try this ρ.
1 ρ
SLIDE 8
Construction of a commutator multiplication
Task
◮ Given: L. ◮ Asked: A multiplication [., .] and
a nilpotency killer ρ.
Finding [., .] and ρ
◮ We try this ρ. ◮ We want a multiplication [., .] with
[1, ρ] = ρ. 1 ρ
SLIDE 9
Construction of a commutator multiplication
Task
◮ Given: L. ◮ Asked: A multiplication [., .] and
a nilpotency killer ρ.
Finding [., .] and ρ
◮ We try this ρ. ◮ We want a multiplication [., .] with
[1, ρ] = ρ.
◮ Not possible because:
1 ρ
SLIDE 10
Construction of a commutator multiplication
Task
◮ Given: L. ◮ Asked: A multiplication [., .] and
a nilpotency killer ρ.
Finding [., .] and ρ
◮ We try this ρ. ◮ We want a multiplication [., .] with
[1, ρ] = ρ.
◮ Not possible because:
[α, ρ] ≤ α ∧ ρ = 0 and 1 α ρ
SLIDE 11
Construction of a commutator multiplication
Task
◮ Given: L. ◮ Asked: A multiplication [., .] and
a nilpotency killer ρ.
Finding [., .] and ρ
◮ We try this ρ. ◮ We want a multiplication [., .] with
[1, ρ] = ρ.
◮ Not possible because:
[α, ρ] ≤ α ∧ ρ = 0 and [β, ρ] ≤ β ∧ ρ = 0 and hence [1, ρ] = [α ∨ β, ρ] = [α, ρ] ∨ [β, ρ] = 0 ∨ 0 = 0. 1 α β ρ
SLIDE 12
Construction of a commutator multiplication
Task
◮ Asked: A multiplication [., .] and
a nilpotency killer ρ.
Finding [., .] and ρ
1
SLIDE 13
Construction of a commutator multiplication
Task
◮ Asked: A multiplication [., .] and
a nilpotency killer ρ.
Finding [., .] and ρ
◮ We try this ρ.
1 ρ
SLIDE 14
Construction of a commutator multiplication
Task
◮ Asked: A multiplication [., .] and
a nilpotency killer ρ.
Finding [., .] and ρ
◮ We try this ρ. ◮ Now we will succeed!
1 ρ
SLIDE 15
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ]. 1 ρ
SLIDE 16
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ]. 1 ρ
SLIDE 17
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ].
◮ Find all meet irreducibles η with
I[η, η+] I[0, ρ] 1 ρ
SLIDE 18
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ].
◮ Find all meet irreducibles η with
I[η, η+] I[0, ρ] 1 ρ
SLIDE 19
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ].
◮ Find all meet irreducibles η with
I[η, η+] I[0, ρ]
◮ Let Γ be their join.
1 ρ
SLIDE 20
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ].
◮ Find all meet irreducibles η with
I[η, η+] I[0, ρ]
◮ Let Γ be their join.
1 Γ ρ
SLIDE 21
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ].
◮ Find all meet irreducibles η with
I[η, η+] I[0, ρ]
◮ Let Γ be their join. ◮ Find all join irreducibles ν with
I[ν−, ν] I[0, ρ]. 1 Γ ρ
SLIDE 22
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ].
◮ Find all meet irreducibles η with
I[η, η+] I[0, ρ]
◮ Let Γ be their join. ◮ Find all join irreducibles ν with
I[ν−, ν] I[0, ρ]. 1 Γ ρ
SLIDE 23
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ].
◮ Find all meet irreducibles η with
I[η, η+] I[0, ρ]
◮ Let Γ be their join. ◮ Find all join irreducibles ν with
I[ν−, ν] I[0, ρ].
◮ Let ∆ be their join.
1 Γ ρ
SLIDE 24
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ].
◮ Find all meet irreducibles η with
I[η, η+] I[0, ρ]
◮ Let Γ be their join. ◮ Find all join irreducibles ν with
I[ν−, ν] I[0, ρ].
◮ Let ∆ be their join.
1 Γ ∆ ρ
SLIDE 25
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ].
◮ Find all meet irreducibles η with
I[η, η+] I[0, ρ]
◮ Let Γ be their join. ◮ Find all join irreducibles ν with
I[ν−, ν] I[0, ρ].
◮ Let ∆ be their join. ◮ Let Θ be the congruence of L
generated by (∆, 1). 1 Γ ∆ ρ
SLIDE 26
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ].
◮ Find all meet irreducibles η with
I[η, η+] I[0, ρ]
◮ Let Γ be their join. ◮ Find all join irreducibles ν with
I[ν−, ν] I[0, ρ].
◮ Let ∆ be their join. ◮ Let Θ be the congruence of L
generated by (∆, 1). 1 Γ ∆ ρ
SLIDE 27
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ].
◮ Find all meet irreducibles η with
I[η, η+] I[0, ρ]
◮ Let Γ be their join. ◮ Find all join irreducibles ν with
I[ν−, ν] I[0, ρ].
◮ Let ∆ be their join. ◮ Let Θ be the congruence of L
generated by (∆, 1).
◮ Define s(x) := {z | (z, x) ∈ Θ}.
1 Γ ∆ ρ
SLIDE 28
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ].
◮ Find all meet irreducibles η with
I[η, η+] I[0, ρ]
◮ Let Γ be their join. ◮ Find all join irreducibles ν with
I[ν−, ν] I[0, ρ].
◮ Let ∆ be their join. ◮ Let Θ be the congruence of L
generated by (∆, 1).
◮ Define s(x) := {z | (z, x) ∈ Θ}.
1 Γ
s(1)
ρ
SLIDE 29
Construction of the multiplication
◮ Find all intervals projective to
I[0, ρ].
◮ Find all meet irreducibles η with
I[η, η+] I[0, ρ]
◮ Let Γ be their join. ◮ Find all join irreducibles ν with
I[ν−, ν] I[0, ρ].
◮ Let ∆ be their join. ◮ Let Θ be the congruence of L
generated by (∆, 1).
◮ Define s(x) := {z | (z, x) ∈ Θ}.
1 Γ ∆ ρ
SLIDE 30
Construction of the multiplication
The multiplication
1 Γ ∆ ρ
SLIDE 31
Construction of the multiplication
The multiplication
◮ Define [x, y] := 0 if x ≤ Γ and
y ≤ Γ. 1 Γ ∆ ρ
SLIDE 32 Construction of the multiplication
The multiplication
◮ Define [x, y] := 0 if x ≤ Γ and
y ≤ Γ.
◮ Define [x, y] := s(x ∧ y)
1 Γ ∆ ρ
SLIDE 33 Construction of the multiplication
The multiplication
◮ Define [x, y] := 0 if x ≤ Γ and
y ≤ Γ.
◮ Define [x, y] := s(x ∧ y)
Properties of the multiplication
◮ [., .] is commutative, below meet.
1 Γ ∆ ρ
SLIDE 34 Construction of the multiplication
The multiplication
◮ Define [x, y] := 0 if x ≤ Γ and
y ≤ Γ.
◮ Define [x, y] := s(x ∧ y)
Properties of the multiplication
◮ [., .] is commutative, below meet. ◮ [., .] is join distributive.
1 Γ ∆ ρ
SLIDE 35 Construction of the multiplication
The multiplication
◮ Define [x, y] := 0 if x ≤ Γ and
y ≤ Γ.
◮ Define [x, y] := s(x ∧ y)
Properties of the multiplication
◮ [., .] is commutative, below meet. ◮ [., .] is join distributive. ◮ We have [1, ρ] = ρ.
1 Γ ∆ ρ
SLIDE 36 Construction of the multiplication
The multiplication
◮ Define [x, y] := 0 if x ≤ Γ and
y ≤ Γ.
◮ Define [x, y] := s(x ∧ y)
Properties of the multiplication
◮ [., .] is commutative, below meet. ◮ [., .] is join distributive. ◮ We have [1, ρ] = ρ. ◮ Conclusion: L does not force
nilpotent type. 1 Γ ∆ ρ
SLIDE 37
Lattices with only nilpotent commutator multiplications
The general content
This construction was possible because there were: 1
SLIDE 38 Lattices with only nilpotent commutator multiplications
The general content
This construction was possible because there were:
◮ a join irreducible ρ ∈ L with
Γ = Γ(ρ−, ρ) < 1, where Γ =
- {η | η m.i. andI[ρ−, ρ] I[η, η+]}.
1 ρ
SLIDE 39 Lattices with only nilpotent commutator multiplications
The general content
This construction was possible because there were:
◮ a join irreducible ρ ∈ L with
Γ = Γ(ρ−, ρ) < 1, where Γ =
- {η | η m.i. andI[ρ−, ρ] I[η, η+]}.
1 Γ ρ
SLIDE 40 Lattices with only nilpotent commutator multiplications
The general content
This construction was possible because there were:
◮ a join irreducible ρ ∈ L with
Γ = Γ(ρ−, ρ) < 1, where Γ =
- {η | η m.i. andI[ρ−, ρ] I[η, η+]}.
1 Γ ρ
Theorem
Let L be a modular lattice of finite height. TFAE:
◮ L allows a non-nilpotent commutator multiplication. ◮ ∃α ≺ β ∈ L such that Γ(α, β) < 1.
SLIDE 41 The largest commutator multiplication
Lemma (Czelakowski)
The join of commutator multiplications is again a commutator
- multiplication. Hence on a given lattice, there is one largest
commutator multiplication. Czelakowski: “The characterization of the operation •Ω in modular algebraic lattices is an open and challenging problem.”
Description of the largest commutator multiplication
Let ⌈., .⌉ denote the largest commutator multiplication on L.
◮ We have no description of ⌈x, y⌉ yet. ◮ We have no description of the associated residuation
(x : y) = {z | ⌈z, y⌉ ≤ x} either.
◮ We can describe (x : y) if x ≺ y!
SLIDE 42 The largest commutator operation
Theorem
Let L be a bialgebraic modular lattice, and let (x : y) be the residuation
- peration associated with the largest
commutator multiplication. Let α, β ∈ L be such that α ≺ β. Then (α : β) = Γ(α, β). Γblue Γred
SLIDE 43
Open problem
Definition
L forces abelian type if [x, y] = 0 is the only commutator multiplication on L.
Problem
Characterize those modular lattices of finite height that force abelian type.
Theorem
Let L be a complete lattice. If L has a complete (0, 1)-sublattice K that is algebraic, modular, simple, complemented, and has at least 3 elements, then L forces abelian type.
SLIDE 44
References
◮ G. Birkhoff, Lattice Theory, AMS, editions 1948 and 1967. ◮ J. Czelakowski, Additivity of the commutator and
residuation, Reports on Mathematical Logic (2008), no. 43, 109–132.
◮ J. Czelakowski, The equationally-defined commutator,
Birkhäuser/Springer, Cham, 2015.
◮ E. Aichinger, Congruence lattices forcing nilpotency, arXiv,
to appear in Journal of Algebra and its Applications.
