Upper and Lower Semimodularity of the Supercharacter Theory Lattices - PowerPoint PPT Presentation

Upper and Lower Semimodularity of the Supercharacter Theory Lattices of Cyclic Groups Samuel Benidt, William Hall, & Anders Hendrickson November 10, 2009

Definitions Our Work Two main theorems Outline Definitions 1 Lattices Groups and subgroups Supercharacter Theories Our Work 2 Goals and Strategy Analysis of Specific Cases Two main theorems 3 Upper Semimodularity Lower Semimodularity Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Lattices As you may know, a partially ordered set is a set with an order that requires the following for elements a , b in the set: either a ≤ b or a ≥ b or a and b are incomparable Definition A lattice is a partially ordered set in which any two elements a and b have a unique least upper bound, a ∨ b , and greatest lower bound, a ∧ b . Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Examples A Lattice Definition We say an element a of a lattice covers another element d if a ≥ d and there are no elements between a and d . We write d � a . Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Examples A Lattice Not a lattice Definition We say an element a of a lattice covers another element d if a ≥ d and there are no elements between a and d . We write d � a . Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Examples A Lattice Not a lattice Definition We say an element a of a lattice covers another element d if a ≥ d and there are no elements between a and d . We write d � a . Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Upper Semimodularity Definition A lattice L of finite length, is said to be upper semimodular if the following condition is satisfied: if a ∧ b � a , b then a , b � a ∨ b . Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Upper Semimodularity Definition A lattice L of finite length, is said to be upper semimodular if the following condition is satisfied: if a ∧ b � a , b then a , b � a ∨ b . ⇒ Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Lower Semimodularity Definition Let L be a lattice of finite length. Then L is lower semimodular if for all a , b ∈ L , if a , b � a ∨ b then a ∧ b � a , b Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Lower Semimodularity Definition Let L be a lattice of finite length. Then L is lower semimodular if for all a , b ∈ L , if a , b � a ∨ b then a ∧ b � a , b ⇒ Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Group Definition A set G , with respect to operation ∗ , is a group if it is associative has an identity (equivalently, contains 1) is closed has inverses We will only consider cyclic groups, in which there is a generator g ∈ G such that each element in the group equals g n for some n ∈ N . Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Group Definition A set G , with respect to operation ∗ , is a group if it is associative has an identity (equivalently, contains 1) is closed has inverses We will only consider cyclic groups, in which there is a generator g ∈ G such that each element in the group equals g n for some n ∈ N . Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Group Definition A set G , with respect to operation ∗ , is a group if it is associative has an identity (equivalently, contains 1) is closed has inverses We will only consider cyclic groups, in which there is a generator g ∈ G such that each element in the group equals g n for some n ∈ N . Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Group Definition A set G , with respect to operation ∗ , is a group if it is associative has an identity (equivalently, contains 1) is closed has inverses We will only consider cyclic groups, in which there is a generator g ∈ G such that each element in the group equals g n for some n ∈ N . Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Subgroup Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z 12 Group Elements 1 , g , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 , g 9 , g 10 , g 11 Z 12 1 , g 2 , g 4 , g 6 , g 8 , g 10 Z 6 1 , g 3 , g 6 , g 9 Z 4 1 , g 4 , g 8 Z 3 1 , g 6 Z 2 1 Z 1 Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Subgroup Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z 12 Group Elements 1 , g , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 , g 9 , g 10 , g 11 Z 12 1 , g 2 , g 4 , g 6 , g 8 , g 10 Z 6 1 , g 3 , g 6 , g 9 Z 4 1 , g 4 , g 8 Z 3 1 , g 6 Z 2 1 Z 1 Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Subgroup Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z 12 Group Elements 1 , g , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 , g 9 , g 10 , g 11 Z 12 1 , g 2 , g 4 , g 6 , g 8 , g 10 Z 6 1 , g 3 , g 6 , g 9 Z 4 1 , g 4 , g 8 Z 3 1 , g 6 Z 2 1 Z 1 Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Subgroup Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z 12 Group Elements 1 , g , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 , g 9 , g 10 , g 11 Z 12 1 , g 2 , g 4 , g 6 , g 8 , g 10 Z 6 1 , g 3 , g 6 , g 9 Z 4 1 , g 4 , g 8 Z 3 1 , g 6 Z 2 1 Z 1 Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Subgroup Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z 12 Group Elements 1 , g , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 , g 9 , g 10 , g 11 Z 12 1 , g 2 , g 4 , g 6 , g 8 , g 10 Z 6 1 , g 3 , g 6 , g 9 Z 4 1 , g 4 , g 8 Z 3 1 , g 6 Z 2 1 Z 1 Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Subgroup Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z 12 Group Elements 1 , g , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 , g 9 , g 10 , g 11 Z 12 1 , g 2 , g 4 , g 6 , g 8 , g 10 Z 6 1 , g 3 , g 6 , g 9 Z 4 1 , g 4 , g 8 Z 3 1 , g 6 Z 2 1 Z 1 Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Subgroup Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z 12 Group Elements 1 , g , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 , g 9 , g 10 , g 11 Z 12 1 , g 2 , g 4 , g 6 , g 8 , g 10 Z 6 1 , g 3 , g 6 , g 9 Z 4 1 , g 4 , g 8 Z 3 1 , g 6 Z 2 1 Z 1 Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Lattices Our Work Groups and subgroups Two main theorems Supercharacter Theories Subgroup Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z 12 Group Elements 1 , g , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 , g 9 , g 10 , g 11 Z 12 1 , g 2 , g 4 , g 6 , g 8 , g 10 Z 6 1 , g 3 , g 6 , g 9 Z 4 1 , g 4 , g 8 Z 3 1 , g 6 Z 2 1 Z 1 Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups