Lattices allowing only nilpotent commutator operations Erhard - PowerPoint PPT Presentation

Lattices allowing only nilpotent commutator operations Erhard Aichinger Institute for Algebra Johannes Kepler University Linz, Austria February 2017, AAA93 Supported by the Austrian Science Fund (FWF) : P24077 and P29931

Does L force nilpotency? 1 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? 0

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 .

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.

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.

Construction of a commutator multiplication 1 Task ◮ Given: L . ◮ Asked: A multiplication [ ., . ] and a nilpotency killer ρ . Finding [ ., . ] and ρ 0

Construction of a commutator multiplication 1 Task ◮ Given: L . ◮ Asked: A multiplication [ ., . ] and a nilpotency killer ρ . Finding [ ., . ] and ρ ◮ We try this ρ . ρ 0

Construction of a commutator multiplication 1 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 . ρ 0

Construction of a commutator multiplication 1 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: ρ 0

Construction of a commutator multiplication 1 Task ◮ Given: L . α ◮ 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: ρ [ α, ρ ] ≤ α ∧ ρ = 0 and 0

Construction of a commutator multiplication 1 Task ◮ Given: L . α β ◮ 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: ρ [ α, ρ ] ≤ α ∧ ρ = 0 and [ β, ρ ] ≤ β ∧ ρ = 0 and hence 0 [ 1 , ρ ] = [ α ∨ β, ρ ] = [ α, ρ ] ∨ [ β, ρ ] = 0 ∨ 0 = 0.

Construction of a commutator multiplication 1 Task ◮ Asked: A multiplication [ ., . ] and a nilpotency killer ρ . Finding [ ., . ] and ρ 0

Construction of a commutator multiplication 1 Task ◮ Asked: A multiplication [ ., . ] and a nilpotency killer ρ . Finding [ ., . ] and ρ ρ ◮ We try this ρ . 0

Construction of a commutator multiplication 1 Task ◮ Asked: A multiplication [ ., . ] and a nilpotency killer ρ . Finding [ ., . ] and ρ ρ ◮ We try this ρ . ◮ Now we will succeed! 0

Construction of the multiplication 1 ◮ Find all intervals projective to I [ 0 , ρ ] . ρ 0

Construction of the multiplication 1 ◮ Find all intervals projective to I [ 0 , ρ ] . ρ 0

Construction of the multiplication 1 ◮ Find all intervals projective to I [ 0 , ρ ] . ◮ Find all meet irreducibles η with I [ η, η + ] � I [ 0 , ρ ] ρ 0

Construction of the multiplication 1 ◮ Find all intervals projective to I [ 0 , ρ ] . ◮ Find all meet irreducibles η with I [ η, η + ] � I [ 0 , ρ ] ρ 0

Construction of the multiplication 1 ◮ 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. ρ 0

Construction of the multiplication 1 Γ ◮ 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. ρ 0

Construction of the multiplication 1 Γ ◮ 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 ] . ρ 0

Construction of the multiplication 1 Γ ◮ 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 ] . ρ 0

Construction of the multiplication 1 Γ ◮ 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 ] . ρ ◮ Let ∆ be their join. 0

Construction of the multiplication 1 Γ ◮ 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 ] . ρ ◮ Let ∆ be their join. 0

Construction of the multiplication 1 Γ ◮ 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 ] . ρ ◮ Let ∆ be their join. ◮ Let Θ be the congruence of L generated by ( ∆ , 1 ) . 0

Construction of the multiplication 1 Γ ◮ 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 ] . ρ ◮ Let ∆ be their join. ◮ Let Θ be the congruence of L generated by ( ∆ , 1 ) . 0

Construction of the multiplication 1 Γ ◮ 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 ] . ρ ◮ Let ∆ be their join. ◮ Let Θ be the congruence of L generated by ( ∆ , 1 ) . 0 ◮ Define s ( x ) := � { z | ( z , x ) ∈ Θ } .

Construction of the multiplication 1 Γ ◮ Find all intervals projective to I [ 0 , ρ ] . s ( 1 ) ◮ 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 ] . ρ ◮ Let ∆ be their join. ◮ Let Θ be the congruence of L generated by ( ∆ , 1 ) . 0 ◮ Define s ( x ) := � { z | ( z , x ) ∈ Θ } .

Construction of the multiplication 1 Γ ◮ 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 ] . ρ ◮ Let ∆ be their join. ◮ Let Θ be the congruence of L generated by ( ∆ , 1 ) . 0 ◮ Define s ( x ) := � { z | ( z , x ) ∈ Θ } .

Construction of the multiplication 1 The multiplication Γ ∆ ρ 0

Construction of the multiplication 1 The multiplication Γ ◮ Define [ x , y ] := 0 if x ≤ Γ and y ≤ Γ . ∆ ρ 0

Construction of the multiplication 1 The multiplication Γ ◮ Define [ x , y ] := 0 if x ≤ Γ and y ≤ Γ . ∆ ◮ Define [ x , y ] := s ( x ∧ y ) otherwise. ρ 0

Construction of the multiplication 1 The multiplication Γ ◮ Define [ x , y ] := 0 if x ≤ Γ and y ≤ Γ . ∆ ◮ Define [ x , y ] := s ( x ∧ y ) otherwise. Properties of the multiplication ρ ◮ [ ., . ] is commutative, below meet. 0

Construction of the multiplication 1 The multiplication Γ ◮ Define [ x , y ] := 0 if x ≤ Γ and y ≤ Γ . ∆ ◮ Define [ x , y ] := s ( x ∧ y ) otherwise. Properties of the multiplication ρ ◮ [ ., . ] is commutative, below meet. ◮ [ ., . ] is join distributive. 0

Construction of the multiplication 1 The multiplication Γ ◮ Define [ x , y ] := 0 if x ≤ Γ and y ≤ Γ . ∆ ◮ Define [ x , y ] := s ( x ∧ y ) otherwise. Properties of the multiplication ρ ◮ [ ., . ] is commutative, below meet. ◮ [ ., . ] is join distributive. ◮ We have [ 1 , ρ ] = ρ . 0