Augmented quasigroups: from group duals to Heyting algebras - - PowerPoint PPT Presentation

Augmented quasigroups: from group duals to Heyting algebras

Jonathan D.H. Smith Iowa State University email: jdhsmith@iastate.edu

https://orion.math.iastate.edu/jdhsmith/

Symmetric monoidal categories

Symmetric monoidal categories

A symmetric monoidal category is a category V

Symmetric monoidal categories

A symmetric monoidal category is a category V with a (tensor product) bifunctor ⊗: V × V → V; (A, B) → A ⊗ B

Symmetric monoidal categories

A symmetric monoidal category is a category V with a (tensor product) bifunctor ⊗: V × V → V; (A, B) → A ⊗ B and a (unit) object 1

Symmetric monoidal categories

A symmetric monoidal category is a category V with a (tensor product) bifunctor ⊗: V × V → V; (A, B) → A ⊗ B and a (unit) object 1 such that (V, ⊗, 1) is a commutative Monoid

Symmetric monoidal categories

A symmetric monoidal category is a category V with a (tensor product) bifunctor ⊗: V × V → V; (A, B) → A ⊗ B and a (unit) object 1 such that (V, ⊗, 1) is a commutative Monoid (up to coherent natural isomorphisms like 1 ⊗ A lA A A ⊗ 1

rA

• , etc.)

Symmetric monoidal categories

A symmetric monoidal category is a category V with a (tensor product) bifunctor ⊗: V × V → V; (A, B) → A ⊗ B and a (unit) object 1 such that (V, ⊗, 1) is a commutative Monoid (up to coherent natural isomorphisms like 1 ⊗ A lA A A ⊗ 1

rA

• , etc.)

Examples:

• (Set, ×, ⊤), (Rel, ×, ⊤) with a one-element set ⊤;

Symmetric monoidal categories

A symmetric monoidal category is a category V with a (tensor product) bifunctor ⊗: V × V → V; (A, B) → A ⊗ B and a (unit) object 1 such that (V, ⊗, 1) is a commutative Monoid (up to coherent natural isomorphisms like 1 ⊗ A lA A A ⊗ 1

rA

• , etc.)

Examples:

• (Set, ×, ⊤), (Rel, ×, ⊤) with a one-element set ⊤;
• Category (P, ⊗, P) of finitely-generated free (semi-)modules
• ver a commutative unital (semi-)ring P;

Symmetric monoidal categories

A symmetric monoidal category is a category V with a (tensor product) bifunctor ⊗: V × V → V; (A, B) → A ⊗ B and a (unit) object 1 such that (V, ⊗, 1) is a commutative Monoid (up to coherent natural isomorphisms like 1 ⊗ A lA A A ⊗ 1

rA

• , etc.)

Examples:

• (Set, ×, ⊤), (Rel, ×, ⊤) with a one-element set ⊤;
• Category (P, ⊗, P) of finitely-generated free (semi-)modules
• ver a commutative unital (semi-)ring P;
• Any entropic variety (V, ⊗, I),

with tensor product ⊗ and I free on one generator;

Symmetric monoidal categories

A symmetric monoidal category is a category V with a (tensor product) bifunctor ⊗: V × V → V; (A, B) → A ⊗ B and a (unit) object 1 such that (V, ⊗, 1) is a commutative Monoid (up to coherent natural isomorphisms like 1 ⊗ A lA A A ⊗ 1

rA

• , etc.)

Examples:

• (Set, ×, ⊤), (Rel, ×, ⊤) with a one-element set ⊤;
• Category (P, ⊗, P) of finitely-generated free (semi-)modules
• ver a commutative unital (semi-)ring P;
• Any entropic variety (V, ⊗, I),

with tensor product ⊗ and I free on one generator;

• Any category (C, +, ⊥) with coproduct + and initial object ⊥.

Symmetric monoidal categories

A symmetric monoidal category is a category V with a (tensor product) bifunctor ⊗: V × V → V; (A, B) → A ⊗ B and a (unit) object 1 such that (V, ⊗, 1) is a commutative Monoid (up to coherent natural isomorphisms like 1 ⊗ A lA A A ⊗ 1

rA

• , etc.)

Examples:

• (Set, ×, ⊤), (Rel, ×, ⊤) with a one-element set ⊤;
• Category (P, ⊗, P) of finitely-generated free (semi-)modules
• ver a commutative unital (semi-)ring P;
• Any entropic variety (V, ⊗, I),

with tensor product ⊗ and I free on one generator;

• Any category (C, +, ⊥) with coproduct + and initial object ⊥.

Magma: (A, ∇: A ⊗ A → A)

Symmetric monoidal categories

A symmetric monoidal category is a category V with a (tensor product) bifunctor ⊗: V × V → V; (A, B) → A ⊗ B and a (unit) object 1 such that (V, ⊗, 1) is a commutative Monoid (up to coherent natural isomorphisms like 1 ⊗ A lA A A ⊗ 1

rA

• , etc.)

Examples:

• (Set, ×, ⊤), (Rel, ×, ⊤) with a one-element set ⊤;
• Category (P, ⊗, P) of finitely-generated free (semi-)modules
• ver a commutative unital (semi-)ring P;
• Any entropic variety (V, ⊗, I),

with tensor product ⊗ and I free on one generator;

• Any category (C, +, ⊥) with coproduct + and initial object ⊥.

Magma: (A, ∇: A ⊗ A → A) Comagma: (A, ∆: A → A ⊗ A)

Compact closed categories

Compact closed categories

Symmetric monoidal categories (V, ⊗, 1) with extra structure:

Compact closed categories

Symmetric monoidal categories (V, ⊗, 1) with extra structure: contravariant duality functor ∗: V → V;

Compact closed categories

Symmetric monoidal categories (V, ⊗, 1) with extra structure: contravariant duality functor ∗: V → V; natural transformation evaluation evA: A ⊗ A∗ → 1;

Compact closed categories

Symmetric monoidal categories (V, ⊗, 1) with extra structure: contravariant duality functor ∗: V → V; natural transformation evaluation evA: A ⊗ A∗ → 1; and natural transformation coevaluation coevA: 1 → A∗ ⊗ A ,

Compact closed categories

Symmetric monoidal categories (V, ⊗, 1) with extra structure: contravariant duality functor ∗: V → V; natural transformation evaluation evA: A ⊗ A∗ → 1; and natural transformation coevaluation coevA: 1 → A∗ ⊗ A , such that the (mutually dual) yanking conditions hold:

Compact closed categories

Symmetric monoidal categories (V, ⊗, 1) with extra structure: contravariant duality functor ∗: V → V; natural transformation evaluation evA: A ⊗ A∗ → 1; and natural transformation coevaluation coevA: 1 → A∗ ⊗ A , such that the (mutually dual) yanking conditions hold: A1A⊗coev

A ⊗ A∗ ⊗ A ev⊗1A A reduces to 1A

Compact closed categories

Symmetric monoidal categories (V, ⊗, 1) with extra structure: contravariant duality functor ∗: V → V; natural transformation evaluation evA: A ⊗ A∗ → 1; and natural transformation coevaluation coevA: 1 → A∗ ⊗ A , such that the (mutually dual) yanking conditions hold: A1A⊗coev

A ⊗ A∗ ⊗ A ev⊗1A A reduces to 1A

and A∗coev⊗1A

A∗ ⊗ A ⊗ A∗ 1A⊗ev A∗ reduces to 1A∗,

Compact closed categories

Symmetric monoidal categories (V, ⊗, 1) with extra structure: contravariant duality functor ∗: V → V; natural transformation evaluation evA: A ⊗ A∗ → 1; and natural transformation coevaluation coevA: 1 → A∗ ⊗ A , such that the (mutually dual) yanking conditions hold: A1A⊗coev

A ⊗ A∗ ⊗ A ev⊗1A A reduces to 1A

and A∗coev⊗1A

A∗ ⊗ A ⊗ A∗ 1A⊗ev A∗ reduces to 1A∗,

for each object A of V.

Pictorial yanking

r−1

A

1A ⊗ coev

ev ⊗ 1A

lA

✲

A

✲

A ⊗

✲

A

✲

A∗

1

✲

A∗

✲

A

1

✲

A

✲

A

✲

A ⊗

✲

A

Pictorial yanking

r−1

A

1A ⊗ coev

ev ⊗ 1A

lA

✲

A

✲

A

✲

A

✛ A ✛ A ✲

A

✲

A

✲

A

✲

A

✲

A

Finite-dimensional real vector spaces

Finite-dimensional real vector spaces

Suppose spaces A, B have respective finite bases X, Y ,

Finite-dimensional real vector spaces

Suppose spaces A, B have respective finite bases X, Y , so A ⊗ B has basis X × Y = {x ⊗ y | x ∈ X , y ∈ Y }.

Finite-dimensional real vector spaces

Suppose spaces A, B have respective finite bases X, Y , so A ⊗ B has basis X × Y = {x ⊗ y | x ∈ X , y ∈ Y }. Unit object 1 = R, with basis {1}.

Finite-dimensional real vector spaces

Suppose spaces A, B have respective finite bases X, Y , so A ⊗ B has basis X × Y = {x ⊗ y | x ∈ X , y ∈ Y }. Unit object 1 = R, with basis {1}. Duality functor ∗: R → R; A → A∗ = R(A, R)

Finite-dimensional real vector spaces

Suppose spaces A, B have respective finite bases X, Y , so A ⊗ B has basis X × Y = {x ⊗ y | x ∈ X , y ∈ Y }. Unit object 1 = R, with basis {1}. Duality functor ∗: R → R; A → A∗ = R(A, R) Evaluation evA: A ⊗ A∗ → R; x′ ⊗ δx → x′δx = δx′,x

Finite-dimensional real vector spaces

Suppose spaces A, B have respective finite bases X, Y , so A ⊗ B has basis X × Y = {x ⊗ y | x ∈ X , y ∈ Y }. Unit object 1 = R, with basis {1}. Duality functor ∗: R → R; A → A∗ = R(A, R) Evaluation evA: A ⊗ A∗ → R; x′ ⊗ δx → x′δx = δx′,x Coevaluation coevA: R → A∗ ⊗ A; 1 → ∑

x∈X δx ⊗ x

Finite-dimensional real vector spaces

Suppose spaces A, B have respective finite bases X, Y , so A ⊗ B has basis X × Y = {x ⊗ y | x ∈ X , y ∈ Y }. Unit object 1 = R, with basis {1}. Duality functor ∗: R → R; A → A∗ = R(A, R) Evaluation evA: A ⊗ A∗ → R; x′ ⊗ δx → x′δx = δx′,x Coevaluation coevA: R → A∗ ⊗ A; 1 → ∑

x∈X δx ⊗ x

First yanking condition x′

✤

r−1

A x′ ⊗ 1

✤

1A⊗coev

x′ ⊗ ∑

x∈X δx ⊗ x ✤ev⊗1A ∑ x∈X x′δx ⊗ x = 1 ⊗ x′ ✤ lA x′

Examples of compact-closed categories

Examples of compact-closed categories

• The category of finite-dimensional Hilbert spaces;

Examples of compact-closed categories

• The category of finite-dimensional Hilbert spaces;
• Categories of finitely-generated free semimodules over a semiring;

Examples of compact-closed categories

• The category of finite-dimensional Hilbert spaces;
• Categories of finitely-generated free semimodules over a semiring;
• Joyal’s category of Conway games;

Examples of compact-closed categories

• The category of finite-dimensional Hilbert spaces;
• Categories of finitely-generated free semimodules over a semiring;
• Joyal’s category of Conway games;
• The category (Rel, ⊗, ⊤) of relations between sets, say ⊤ = {0},

Examples of compact-closed categories

• The category of finite-dimensional Hilbert spaces;
• Categories of finitely-generated free semimodules over a semiring;
• Joyal’s category of Conway games;
• The category (Rel, ⊗, ⊤) of relations between sets, say ⊤ = {0},
• tensor product A ⊗ B is the Cartesian product,

Examples of compact-closed categories

• The category of finite-dimensional Hilbert spaces;
• Categories of finitely-generated free semimodules over a semiring;
• Joyal’s category of Conway games;
• The category (Rel, ⊗, ⊤) of relations between sets, say ⊤ = {0},
• tensor product A ⊗ B is the Cartesian product,
• biproduct A ⊕ B is the disjoint union,

Examples of compact-closed categories

• The category of finite-dimensional Hilbert spaces;
• Categories of finitely-generated free semimodules over a semiring;
• Joyal’s category of Conway games;
• The category (Rel, ⊗, ⊤) of relations between sets, say ⊤ = {0},
• tensor product A ⊗ B is the Cartesian product,
• biproduct A ⊕ B is the disjoint union,
• dual A∗ = A,

Examples of compact-closed categories

• The category of finite-dimensional Hilbert spaces;
• Categories of finitely-generated free semimodules over a semiring;
• Joyal’s category of Conway games;
• The category (Rel, ⊗, ⊤) of relations between sets, say ⊤ = {0},
• tensor product A ⊗ B is the Cartesian product,
• biproduct A ⊕ B is the disjoint union,
• dual A∗ = A,
• evA = {(a ⊗ a, 0) | a ∈ A},

Examples of compact-closed categories

• The category of finite-dimensional Hilbert spaces;
• Categories of finitely-generated free semimodules over a semiring;
• Joyal’s category of Conway games;
• The category (Rel, ⊗, ⊤) of relations between sets, say ⊤ = {0},
• tensor product A ⊗ B is the Cartesian product,
• biproduct A ⊕ B is the disjoint union,
• dual A∗ = A,
• evA = {(a ⊗ a, 0) | a ∈ A},
• coevA = {(0, a ⊗ a) | a ∈ A},

Examples of compact-closed categories

• The category of finite-dimensional Hilbert spaces;
• Categories of finitely-generated free semimodules over a semiring;
• Joyal’s category of Conway games;
• The category (Rel, ⊗, ⊤) of relations between sets, say ⊤ = {0},
• tensor product A ⊗ B is the Cartesian product,
• biproduct A ⊕ B is the disjoint union,
• dual A∗ = A,
• evA = {(a ⊗ a, 0) | a ∈ A},
• coevA = {(0, a ⊗ a) | a ∈ A},
• yanking {(a, a ⊗ b ⊗ b) | a, b ∈ A} ◦ {(a ⊗ a ⊗ b, b) | a, b ∈ A} =

A.

Augmented magmas

Augmented magmas

Augmented magma: (A, µ, ∆, ε) in compact closed (V, ⊗, 1) with:

Augmented magmas

Augmented magma: (A, µ, ∆, ε) in compact closed (V, ⊗, 1) with: multiplication (structure) µ: A ⊗ A → A∗,

Augmented magmas

Augmented magma: (A, µ, ∆, ε) in compact closed (V, ⊗, 1) with: multiplication (structure) µ: A ⊗ A → A∗, comultiplication ∆: A → A ⊗ A,

Augmented magmas

Augmented magma: (A, µ, ∆, ε) in compact closed (V, ⊗, 1) with: multiplication (structure) µ: A ⊗ A → A∗, comultiplication ∆: A → A ⊗ A, and augmentation ε: A → 1,

Augmented magmas

Augmented magma: (A, µ, ∆, ε) in compact closed (V, ⊗, 1) with: multiplication (structure) µ: A ⊗ A → A∗, comultiplication ∆: A → A ⊗ A, and augmentation ε: A → 1, such that A ⊗ A

coevA⊗µ

• ε⊗ε
• A∗ ⊗ A ⊗ A∗

1A∗⊗∆⊗1A∗

A∗ ⊗ A ⊗ A ⊗ A∗

τ⊗evA

• 1

A ⊗ A∗

evA

• commutes.

Group algebras as augmented magmas

Group algebras as augmented magmas

Commutative, unital ring R, finite group G, group algebra RG.

Group algebras as augmented magmas

Commutative, unital ring R, finite group G, group algebra RG. Hopf algebra (RG, ∇, η, ∆, ε, S) with ∆: g → g ⊗ g and ε: g → 1

Group algebras as augmented magmas

Commutative, unital ring R, finite group G, group algebra RG. Hopf algebra (RG, ∇, η, ∆, ε, S) with ∆: g → g ⊗ g and ε: g → 1 gives augmented magma (RG, µ, ∆, ε) in (R, ⊗, R) with multiplication structure µ: RG ⊗ RG → RG∗; g ⊗ h → [δgh: x → δx,gh] .

Group algebras as augmented magmas

Commutative, unital ring R, finite group G, group algebra RG. Hopf algebra (RG, ∇, η, ∆, ε, S) with ∆: g → g ⊗ g and ε: g → 1 gives augmented magma (RG, µ, ∆, ε) in (R, ⊗, R) with multiplication structure µ: RG ⊗ RG → RG∗; g ⊗ h → [δgh: x → δx,gh] . Diagram chase for the augmented magma condition: g ⊗ h ✤

coevA⊗µ

• ❴

ε⊗ε

• ∑

x∈G δx ⊗ x ⊗ δgh

✤

1A∗⊗∆⊗1A∗

∑

x∈G δx ⊗ x ⊗ x ⊗ δgh

❴

τ⊗evA

• 1 = δgh,gh

∑

x∈G δx,gh(x ⊗ δx)

✤

evA

Group algebras as augmented magmas

Commutative, unital ring R, finite group G, group algebra RG. Hopf algebra (RG, ∇, η, ∆, ε, S) with ∆: g → g ⊗ g and ε: g → 1 gives augmented magma (RG, µ, ∆, ε) in (R, ⊗, R) with multiplication structure µ: RG ⊗ RG → RG∗; g ⊗ h → [δgh: x → δx,gh] . Diagram chase for the augmented magma condition: g ⊗ h ✤

coevA⊗µ

• ❴

ε⊗ε

• ∑

x∈G δx ⊗ x ⊗ δgh

✤

1A∗⊗∆⊗1A∗

∑

x∈G δx ⊗ x ⊗ x ⊗ δgh

❴

τ⊗evA

• 1 = δgh,gh

∑

x∈G δx,gh(x ⊗ δx)

✤

evA

• Remark: If R = Z, then ε: g → 1 is the augmentation in ZG.

Group algebras as augmented magmas

Commutative, unital ring R, finite group G, group algebra RG. Hopf algebra (RG, ∇, η, ∆, ε, S) with ∆: g → g ⊗ g and ε: g → 1 gives augmented magma (RG, µ, ∆, ε) in (R, ⊗, R) with multiplication structure µ: RG ⊗ RG → RG∗; g ⊗ h → [δgh: x → δx,gh] . Diagram chase for the augmented magma condition: g ⊗ h ✤

coevA⊗µ

• ❴

ε⊗ε

• ∑

x∈G δx ⊗ x ⊗ δgh

✤

1A∗⊗∆⊗1A∗

∑

x∈G δx ⊗ x ⊗ x ⊗ δgh

❴

τ⊗evA

• 1 = δgh,gh

∑

x∈G δx,gh(x ⊗ δx)

✤

evA

• Remark: If R = Z, then ε: g → 1 is the augmentation in ZG.

In general, the augmentation need not be a counit for ∆.

Hypermagmas

Hypermagmas

Consider set A with function A × A → 2A; (x, y) → x ⋄ y.

Hypermagmas

Consider set A with function A × A → 2A; (x, y) → x ⋄ y. In (Rel, ⊗, ⊤), take augmentation ε = {(x, 0) | x ∈ A},

Hypermagmas

Consider set A with function A × A → 2A; (x, y) → x ⋄ y. In (Rel, ⊗, ⊤), take augmentation ε = {(x, 0) | x ∈ A}, comultiplication ∆ = {(x, x ⊗ x) | x ∈ A}, i.e., diagonal relation,

Hypermagmas

Consider set A with function A × A → 2A; (x, y) → x ⋄ y. In (Rel, ⊗, ⊤), take augmentation ε = {(x, 0) | x ∈ A}, comultiplication ∆ = {(x, x ⊗ x) | x ∈ A}, i.e., diagonal relation, and multiplication relation {(x ⊗ y, z) | x, y, z ∈ A, z ∈ x ⋄ y}.

Hypermagmas

Consider set A with function A × A → 2A; (x, y) → x ⋄ y. In (Rel, ⊗, ⊤), take augmentation ε = {(x, 0) | x ∈ A}, comultiplication ∆ = {(x, x ⊗ x) | x ∈ A}, i.e., diagonal relation, and multiplication relation {(x ⊗ y, z) | x, y, z ∈ A, z ∈ x ⋄ y}. Hypermagma: x ⋄ y is nonempty for all x, y in A.

Hypermagmas

Consider set A with function A × A → 2A; (x, y) → x ⋄ y. In (Rel, ⊗, ⊤), take augmentation ε = {(x, 0) | x ∈ A}, comultiplication ∆ = {(x, x ⊗ x) | x ∈ A}, i.e., diagonal relation, and multiplication relation {(x ⊗ y, z) | x, y, z ∈ A, z ∈ x ⋄ y}. Hypermagma: x ⋄ y is nonempty for all x, y in A. Theorem: Set A with function A × A → 2A; (x, y) → x ⋄ y

Hypermagmas

Consider set A with function A × A → 2A; (x, y) → x ⋄ y. In (Rel, ⊗, ⊤), take augmentation ε = {(x, 0) | x ∈ A}, comultiplication ∆ = {(x, x ⊗ x) | x ∈ A}, i.e., diagonal relation, and multiplication relation {(x ⊗ y, z) | x, y, z ∈ A, z ∈ x ⋄ y}. Hypermagma: x ⋄ y is nonempty for all x, y in A. Theorem: Set A with function A × A → 2A; (x, y) → x ⋄ y forms a hypermagma if and only if (A, µ, ∆, ε) is an augmented magma in the category (Rel, ⊗, ⊤).

Hypermagmas

Consider set A with function A × A → 2A; (x, y) → x ⋄ y. In (Rel, ⊗, ⊤), take augmentation ε = {(x, 0) | x ∈ A}, comultiplication ∆ = {(x, x ⊗ x) | x ∈ A}, i.e., diagonal relation, and multiplication relation {(x ⊗ y, z) | x, y, z ∈ A, z ∈ x ⋄ y}. Hypermagma: x ⋄ y is nonempty for all x, y in A. Theorem: Set A with function A × A → 2A; (x, y) → x ⋄ y forms a hypermagma if and only if (A, µ, ∆, ε) is an augmented magma in the category (Rel, ⊗, ⊤). Magmas and hypermagmas treated uniformly, regardless of type!

Hypermagmas

Consider set A with function A × A → 2A; (x, y) → x ⋄ y. In (Rel, ⊗, ⊤), take augmentation ε = {(x, 0) | x ∈ A}, comultiplication ∆ = {(x, x ⊗ x) | x ∈ A}, i.e., diagonal relation, and multiplication relation {(x ⊗ y, z) | x, y, z ∈ A, z ∈ x ⋄ y}. Hypermagma: x ⋄ y is nonempty for all x, y in A. Theorem: Set A with function A × A → 2A; (x, y) → x ⋄ y forms a hypermagma if and only if (A, µ, ∆, ε) is an augmented magma in the category (Rel, ⊗, ⊤). Magmas and hypermagmas treated uniformly, regardless of type! In the magma case, (A, µ, ∆, ε) lies in (Set, ⊗, ⊤).

Currying and braiding in compact closed categories

Currying and braiding in compact closed categories

Compact closed category (V, ⊗, 1).

Currying and braiding in compact closed categories

Compact closed category (V, ⊗, 1). Lemma: There is a natural isomorphism with components φA,B,C : V(B ⊗ A, C) → V(B, C ⊗ A∗) at objects A, B, C of V.

Currying and braiding in compact closed categories

Compact closed category (V, ⊗, 1). Lemma: There is a natural isomorphism with components φA,B,C : V(B ⊗ A, C) → V(B, C ⊗ A∗) at objects A, B, C of V. For an object A of V, define τ13: A3 ⊗ A2 ⊗ A1 → A1 ⊗ A2 ⊗ A3; a3 ⊗ a2 ⊗ a1 → a1 ⊗ a2 ⊗ a3

Currying and braiding in compact closed categories

Compact closed category (V, ⊗, 1). Lemma: There is a natural isomorphism with components φA,B,C : V(B ⊗ A, C) → V(B, C ⊗ A∗) at objects A, B, C of V. For an object A of V, define τ13: A3 ⊗ A2 ⊗ A1 → A1 ⊗ A2 ⊗ A3; a3 ⊗ a2 ⊗ a1 → a1 ⊗ a2 ⊗ a3 and τ23: A1 ⊗ A3 ⊗ A2 → A1 ⊗ A2 ⊗ A3; a1 ⊗ a3 ⊗ a2 → a1 ⊗ a2 ⊗ a3

Augmented quasigroups

Augmented quasigroups

Given an augmented magma (A, µ, ∆, ε) in (V, ⊗, 1),

Augmented quasigroups

Given an augmented magma (A, µ, ∆, ε) in (V, ⊗, 1), have right division (structure) ρ: A ⊗ A → A∗ with ρ = µφ−1

A,A⊗A,1τ∗ 13φA,A⊗A,1

Augmented quasigroups

Given an augmented magma (A, µ, ∆, ε) in (V, ⊗, 1), have right division (structure) ρ: A ⊗ A → A∗ with ρ = µφ−1

A,A⊗A,1τ∗ 13φA,A⊗A,1

and left division (structure) λ: A ⊗ A → A∗ with λ = µφ−1

A,A⊗A,1τ∗ 23φA,A⊗A,1.

Augmented quasigroups

Given an augmented magma (A, µ, ∆, ε) in (V, ⊗, 1), have right division (structure) ρ: A ⊗ A → A∗ with ρ = µφ−1

A,A⊗A,1τ∗ 13φA,A⊗A,1

and left division (structure) λ: A ⊗ A → A∗ with λ = µφ−1

A,A⊗A,1τ∗ 23φA,A⊗A,1.

(A, µ, ρ, λ, ∆, ε) is the (augmented) prequasigroup on (A, µ, ∆, ε).

Augmented quasigroups

Given an augmented magma (A, µ, ∆, ε) in (V, ⊗, 1), have right division (structure) ρ: A ⊗ A → A∗ with ρ = µφ−1

A,A⊗A,1τ∗ 13φA,A⊗A,1

and left division (structure) λ: A ⊗ A → A∗ with λ = µφ−1

A,A⊗A,1τ∗ 23φA,A⊗A,1.

(A, µ, ρ, λ, ∆, ε) is the (augmented) prequasigroup on (A, µ, ∆, ε). Augmented quasigroup: Augmented magma (A, µ, ∆, ε) for which (A, ρ, ∆, ε) and (A, λ, ∆, ε) are augmented magmas.

(Quasi-)Group algebras as augmented quasigroups

(Quasi-)Group algebras as augmented quasigroups

Group algebra RG had multiplication structure µ: RG ⊗ RG → RG∗; x ⊗ y → [δxy : z → δz,xy].

(Quasi-)Group algebras as augmented quasigroups

Group algebra RG had multiplication structure µ: RG ⊗ RG → RG∗; x ⊗ y → [δxy : z → δz,xy]. Thus µφ−1

RG,RG⊗RG,R: x ⊗ y ⊗ z → δz,xy,

(Quasi-)Group algebras as augmented quasigroups

Group algebra RG had multiplication structure µ: RG ⊗ RG → RG∗; x ⊗ y → [δxy : z → δz,xy]. Thus µφ−1

RG,RG⊗RG,R: x ⊗ y ⊗ z → δz,xy,

whence µ φ−1

RG,RG⊗RG,R τ∗ 13: z ⊗ y ⊗ x → δz,xy = δx,zy−1 = δx,z/y,

(Quasi-)Group algebras as augmented quasigroups

Group algebra RG had multiplication structure µ: RG ⊗ RG → RG∗; x ⊗ y → [δxy : z → δz,xy]. Thus µφ−1

RG,RG⊗RG,R: x ⊗ y ⊗ z → δz,xy,

whence µ φ−1

RG,RG⊗RG,R τ∗ 13: z ⊗ y ⊗ x → δz,xy = δx,zy−1 = δx,z/y,

so right division µ φ−1

RG,RG⊗RG,R τ∗ 13 φRG,RG⊗RG,R = ρ: z ⊗ y → δz/y.

(Quasi-)Group algebras as augmented quasigroups

Group algebra RG had multiplication structure µ: RG ⊗ RG → RG∗; x ⊗ y → [δxy : z → δz,xy]. Thus µφ−1

RG,RG⊗RG,R: x ⊗ y ⊗ z → δz,xy,

whence µ φ−1

RG,RG⊗RG,R τ∗ 13: z ⊗ y ⊗ x → δz,xy = δx,zy−1 = δx,z/y,

so right division µ φ−1

RG,RG⊗RG,R τ∗ 13 φRG,RG⊗RG,R = ρ: z ⊗ y → δz/y.

Similarly, have left division structure λ: x ⊗ z → δx−1z = δx\z.

(Quasi-)Group algebras as augmented quasigroups

Group algebra RG had multiplication structure µ: RG ⊗ RG → RG∗; x ⊗ y → [δxy : z → δz,xy]. Thus µφ−1

RG,RG⊗RG,R: x ⊗ y ⊗ z → δz,xy,

whence µ φ−1

RG,RG⊗RG,R τ∗ 13: z ⊗ y ⊗ x → δz,xy = δx,zy−1 = δx,z/y,

so right division µ φ−1

RG,RG⊗RG,R τ∗ 13 φRG,RG⊗RG,R = ρ: z ⊗ y → δz/y.

Similarly, have left division structure λ: x ⊗ z → δx−1z = δx\z. Associativity not used for the augmented magma condition on µ,

(Quasi-)Group algebras as augmented quasigroups

Group algebra RG had multiplication structure µ: RG ⊗ RG → RG∗; x ⊗ y → [δxy : z → δz,xy]. Thus µφ−1

RG,RG⊗RG,R: x ⊗ y ⊗ z → δz,xy,

whence µ φ−1

RG,RG⊗RG,R τ∗ 13: z ⊗ y ⊗ x → δz,xy = δx,zy−1 = δx,z/y,

so right division µ φ−1

RG,RG⊗RG,R τ∗ 13 φRG,RG⊗RG,R = ρ: z ⊗ y → δz/y.

Similarly, have left division structure λ: x ⊗ z → δx−1z = δx\z. Associativity not used for the augmented magma condition on µ, so conclude that (RG, µ, ∆, ε) is an augmented quasigroup.

(Quasi-)Group algebras as augmented quasigroups

Group algebra RG had multiplication structure µ: RG ⊗ RG → RG∗; x ⊗ y → [δxy : z → δz,xy]. Thus µφ−1

RG,RG⊗RG,R: x ⊗ y ⊗ z → δz,xy,

whence µ φ−1

RG,RG⊗RG,R τ∗ 13: z ⊗ y ⊗ x → δz,xy = δx,zy−1 = δx,z/y,

so right division µ φ−1

RG,RG⊗RG,R τ∗ 13 φRG,RG⊗RG,R = ρ: z ⊗ y → δz/y.

Similarly, have left division structure λ: x ⊗ z → δx−1z = δx\z. Associativity not used for the augmented magma condition on µ, so conclude that (RG, µ, ∆, ε) is an augmented quasigroup. Works equally well for a finite quasigroup (G, ·, /, \).

Marty quasigroups as augmented quasigroups

Marty quasigroups as augmented quasigroups

(A, ⋄, ⋌, ⋋) with hypermagma structures (A, ⋄), (A, ⋌), and (A, ⋋) is a Marty quasigroup iff ∀ x, y, z ∈ A , z ∈ x ⋄ y ⇔ x ∈ z ⋌ y ⇔ y ∈ x ⋋ z.

Marty quasigroups as augmented quasigroups

(A, ⋄, ⋌, ⋋) with hypermagma structures (A, ⋄), (A, ⋌), and (A, ⋋) is a Marty quasigroup iff ∀ x, y, z ∈ A , z ∈ x ⋄ y ⇔ x ∈ z ⋌ y ⇔ y ∈ x ⋋ z. Hypergroup if ⋄ is associative [F. Marty, 1936].

Marty quasigroups as augmented quasigroups

(A, ⋄, ⋌, ⋋) with hypermagma structures (A, ⋄), (A, ⋌), and (A, ⋋) is a Marty quasigroup iff ∀ x, y, z ∈ A , z ∈ x ⋄ y ⇔ x ∈ z ⋌ y ⇔ y ∈ x ⋋ z. Hypergroup if ⋄ is associative [F. Marty, 1936]. Theorem: Marty quasigroups ≡ augmented quasigroups in (Rel, ⊗, ⊤).

Marty quasigroups as augmented quasigroups

(A, ⋄, ⋌, ⋋) with hypermagma structures (A, ⋄), (A, ⋌), and (A, ⋋) is a Marty quasigroup iff ∀ x, y, z ∈ A , z ∈ x ⋄ y ⇔ x ∈ z ⋌ y ⇔ y ∈ x ⋋ z. Hypergroup if ⋄ is associative [F. Marty, 1936]. Theorem: Marty quasigroups ≡ augmented quasigroups in (Rel, ⊗, ⊤). Corollary: Heyting algebra (A, ∧, →), meet semilattice with x ∧ y ≤ z ⇔ x ≤ y → z ⇔ y ≤ x → z,

Marty quasigroups as augmented quasigroups

(A, ⋄, ⋌, ⋋) with hypermagma structures (A, ⋄), (A, ⋌), and (A, ⋋) is a Marty quasigroup iff ∀ x, y, z ∈ A , z ∈ x ⋄ y ⇔ x ∈ z ⋌ y ⇔ y ∈ x ⋋ z. Hypergroup if ⋄ is associative [F. Marty, 1936]. Theorem: Marty quasigroups ≡ augmented quasigroups in (Rel, ⊗, ⊤). Corollary: Heyting algebra (A, ∧, →), meet semilattice with x ∧ y ≤ z ⇔ x ≤ y → z ⇔ y ≤ x → z, is a Marty quasigroup or augmented quasigroup in (Rel, ⊗, ⊤)

Marty quasigroups as augmented quasigroups

(A, ⋄, ⋌, ⋋) with hypermagma structures (A, ⋄), (A, ⋌), and (A, ⋋) is a Marty quasigroup iff ∀ x, y, z ∈ A , z ∈ x ⋄ y ⇔ x ∈ z ⋌ y ⇔ y ∈ x ⋋ z. Hypergroup if ⋄ is associative [F. Marty, 1936]. Theorem: Marty quasigroups ≡ augmented quasigroups in (Rel, ⊗, ⊤). Corollary: Heyting algebra (A, ∧, →), meet semilattice with x ∧ y ≤ z ⇔ x ≤ y → z ⇔ y ≤ x → z, is a Marty quasigroup or augmented quasigroup in (Rel, ⊗, ⊤) with x ⋄ y = ↑(x ∧ y) , z ⋌ y = ↓(y → z) , x ⋋ z = ↓(x → z).

Multisets as augmented comagmas

Multisets as augmented comagmas

For A = NX in (N, ⊗, N), augmented comagma (A, ∆, ε) with diagonal ∆: x → x ⊗ x and augmentation ε: A → N.

Multisets as augmented comagmas

For A = NX in (N, ⊗, N), augmented comagma (A, ∆, ε) with diagonal ∆: x → x ⊗ x and augmentation ε: A → N. Set A0 = {a ∈ A | a∆ = a ⊗ a and aε ̸= 0} of grouplike elements.

Multisets as augmented comagmas

For A = NX in (N, ⊗, N), augmented comagma (A, ∆, ε) with diagonal ∆: x → x ⊗ x and augmentation ε: A → N. Set A0 = {a ∈ A | a∆ = a ⊗ a and aε ̸= 0} of grouplike elements. Augmented comagma (A, ∆, ε) is multisetlike if A = NA0.

Multisets as augmented comagmas

For A = NX in (N, ⊗, N), augmented comagma (A, ∆, ε) with diagonal ∆: x → x ⊗ x and augmentation ε: A → N. Set A0 = {a ∈ A | a∆ = a ⊗ a and aε ̸= 0} of grouplike elements. Augmented comagma (A, ∆, ε) is multisetlike if A = NA0. [Note: if A ̸= {0} and ε = 0, then (A, ∆, ε) is not multisetlike.]

Multisets as augmented comagmas

For A = NX in (N, ⊗, N), augmented comagma (A, ∆, ε) with diagonal ∆: x → x ⊗ x and augmentation ε: A → N. Set A0 = {a ∈ A | a∆ = a ⊗ a and aε ̸= 0} of grouplike elements. Augmented comagma (A, ∆, ε) is multisetlike if A = NA0. [Note: if A ̸= {0} and ε = 0, then (A, ∆, ε) is not multisetlike.] Then have multiset ε: X → N+; x → w(x),

Multisets as augmented comagmas

For A = NX in (N, ⊗, N), augmented comagma (A, ∆, ε) with diagonal ∆: x → x ⊗ x and augmentation ε: A → N. Set A0 = {a ∈ A | a∆ = a ⊗ a and aε ̸= 0} of grouplike elements. Augmented comagma (A, ∆, ε) is multisetlike if A = NA0. [Note: if A ̸= {0} and ε = 0, then (A, ∆, ε) is not multisetlike.] Then have multiset ε: X → N+; x → w(x), setlike if ε: X → {1}.

Multisets as augmented comagmas

For A = NX in (N, ⊗, N), augmented comagma (A, ∆, ε) with diagonal ∆: x → x ⊗ x and augmentation ε: A → N. Set A0 = {a ∈ A | a∆ = a ⊗ a and aε ̸= 0} of grouplike elements. Augmented comagma (A, ∆, ε) is multisetlike if A = NA0. [Note: if A ̸= {0} and ε = 0, then (A, ∆, ε) is not multisetlike.] Then have multiset ε: X → N+; x → w(x), setlike if ε: X → {1}. Tare weight |X|, gross weight ∑

x∈X w(x).

The Lifting Theorem

The Lifting Theorem

In (N, ⊗, N), multisetlike object (NX, ∆, εX) lifts to, or is covered by, setlike object (NQ, ∆, εQ) if there is a surjective covering function f : Q → X with εX = ∑

x∈X

( ∑

q∈Q qfδx

)

δx.

The Lifting Theorem

In (N, ⊗, N), multisetlike object (NX, ∆, εX) lifts to, or is covered by, setlike object (NQ, ∆, εQ) if there is a surjective covering function f : Q → X with εX = ∑

x∈X

( ∑

q∈Q qfδx

)

δx. In other words, εX : X → N+; x → |f−1{x}|.

The Lifting Theorem

In (N, ⊗, N), multisetlike object (NX, ∆, εX) lifts to, or is covered by, setlike object (NQ, ∆, εQ) if there is a surjective covering function f : Q → X with εX = ∑

x∈X

( ∑

q∈Q qfδx

)

δx. In other words, εX : X → N+; x → |f−1{x}|. Lifting Theorem [Hilton, Wojciechowski (1993); S. (2018)]:

The Lifting Theorem

In (N, ⊗, N), multisetlike object (NX, ∆, εX) lifts to, or is covered by, setlike object (NQ, ∆, εQ) if there is a surjective covering function f : Q → X with εX = ∑

x∈X

( ∑

q∈Q qfδx

)

δx. In other words, εX : X → N+; x → |f−1{x}|. Lifting Theorem [Hilton, Wojciechowski (1993); S. (2018)]: A multisetlike augmented quasigroup (NX, µX, ∆X, εX)

• f gross weight W in (N, ⊗, N)

The Lifting Theorem

In (N, ⊗, N), multisetlike object (NX, ∆, εX) lifts to, or is covered by, setlike object (NQ, ∆, εQ) if there is a surjective covering function f : Q → X with εX = ∑

x∈X

( ∑

q∈Q qfδx

)

δx. In other words, εX : X → N+; x → |f−1{x}|. Lifting Theorem [Hilton, Wojciechowski (1993); S. (2018)]: A multisetlike augmented quasigroup (NX, µX, ∆X, εX)

• f gross weight W in (N, ⊗, N)

lifts to a setlike quasigroup algebra (NQ, µQ, ∆Q, εQ) with |Q| = W.

The dual group of a finite abelian group G . . .

The dual group of a finite abelian group G . . .

. . . is the group G of homomorphisms χ: G → S1 from G to the circle group S1 = {z ∈ C | zz = 1}.

The dual group of a finite abelian group G . . .

. . . is the group G of homomorphisms χ: G → S1 from G to the circle group S1 = {z ∈ C | zz = 1}. Example: Group C3 with character table C3 1 2 χ0 1 1 1 χ1 1 ω ω2 χ2 1 ω2 ω

The dual group of a finite abelian group G . . .

. . . is the group G of homomorphisms χ: G → S1 from G to the circle group S1 = {z ∈ C | zz = 1}. Example: Group C3 with character table C3 1 2 χ0 1 1 1 χ1 1 ω ω2 χ2 1 ω2 ω ⊗ χ0 χ1 χ2 χ0 χ0 χ1 χ2 χ1 χ1 χ2 χ0 χ2 χ2 χ0 χ1

The dual group of a finite abelian group G . . .

. . . is the group G of homomorphisms χ: G → S1 from G to the circle group S1 = {z ∈ C | zz = 1}. Example: Group C3 with character table C3 1 2 χ0 1 1 1 χ1 1 ω ω2 χ2 1 ω2 ω

• G

χ0 χ1 χ2 χ0 χ0 χ1 χ2 χ1 χ1 χ2 χ0 χ2 χ2 χ0 χ1 → ⊗ χ0 χ1 χ2 χ0 χ0 χ1 χ2 χ1 χ1 χ2 χ0 χ2 χ2 χ0 χ1

A dual quasigroup of a finite group G . . .

A dual quasigroup of a finite group G . . .

. . . is a quasigroup lift G of the group’s character algebra NG∨ with εG∨ : χi → χi(1)2 and µG∨ : χi ⊗ χj → [χk → χi(1)χj(1)χk(1)⟨χi ⊗ χj | χk⟩]

A dual quasigroup of a finite group G . . .

. . . is a quasigroup lift G of the group’s character algebra NG∨ with εG∨ : χi → χi(1)2 and µG∨ : χi ⊗ χj → [χk → χi(1)χj(1)χk(1)⟨χi ⊗ χj | χk⟩] Example: Group S3 with character table S3 1 t c χ1 1 1 1 χ2 1 −1 1 θ 2 −1

A dual quasigroup of a finite group G . . .

. . . is a quasigroup lift G of the group’s character algebra NG∨ with εG∨ : χi → χi(1)2 and µG∨ : χi ⊗ χj → [χk → χi(1)χj(1)χk(1)⟨χi ⊗ χj | χk⟩] Example: Group S3 with character table S3 1 t c χ1 1 1 1 χ2 1 −1 1 θ 2 −1 ⊗ χ1 χ2 θ χ1 χ1 χ2 θ χ2 χ2 χ1 θ θ θ θ χ1 + χ2 + θ

A dual quasigroup of a finite group G . . .

. . . is a quasigroup lift G of the group’s character algebra NG∨ with εG∨ : χi → χi(1)2 and µG∨ : χi ⊗ χj → [χk → χi(1)χj(1)χk(1)⟨χi ⊗ χj | χk⟩] Example: Group S3 with character table S3 1 t c χ1 1 1 1 χ2 1 −1 1 θ 2 −1

• S3

χ1 χ2 θ1 θ2 θ3 θ4 χ1 χ1 χ2 θ1 θ2 θ3 θ4 χ2 χ2 χ1 θ2 θ1 θ4 θ3 θ1 θ1 θ2 θ3 θ4 χ1 χ2 θ2 θ2 θ1 θ4 θ3 χ2 χ1 θ3 θ3 θ4 χ1 χ2 θ1 θ2 θ4 θ4 θ3 χ2 χ1 θ2 θ1 → ⊗ χ1 χ2 θ χ1 χ1 χ2 θ χ2 χ2 χ1 θ θ θ θ χ1 + χ2 + θ

A dual quasigroup of a finite group G . . .

. . . is a quasigroup lift G of the group’s character algebra NG∨ with εG∨ : χi → χi(1)2 and µG∨ : χi ⊗ χj → [χk → χi(1)χj(1)χk(1)⟨χi ⊗ χj | χk⟩] Example: Group S3 with character table S3 1 t c χ1 1 1 1 χ2 1 −1 1 θ 2 −1

• S3

χ1 χ2 θ1 θ2 θ3 θ4 χ1 χ1 χ2 θ1 θ2 θ3 θ4 χ2 χ2 χ1 θ2 θ1 θ4 θ3 θ1 θ1 θ2 θ3 θ4 χ1 χ2 θ2 θ2 θ1 θ4 θ3 χ2 χ1 θ3 θ3 θ4 χ1 χ2 θ1 θ2 θ4 θ4 θ3 χ2 χ1 θ2 θ1 → ⊗ χ1 χ2 θ χ1 χ1 χ2 θ χ2 χ2 χ1 θ θ θ θ χ1 + χ2 + θ E.g., θ(1)3⟨θ ⊗ θ|θ⟩ = 8 = |{θ1, θ2}2 ∪ {θ3, θ4}2|.

Thank you for your attention!