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

✤ ✤ 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 ev A : A ⊗ A ∗ → R ; x ′ ⊗ δ x �→ x ′ δ x = δ x ′ ,x Coevaluation coev A : R → A ∗ ⊗ A ; 1 �→ ∑ x ∈ X δ x ⊗ x First yanking condition r − 1 � x ′ ⊗ ∑ x ∈ X δ x ⊗ x ✤ ev ⊗ 1 A � ∑ 1 A ⊗ coev x ∈ X x ′ δ x ⊗ x = 1 ⊗ x ′ ✤ l A � x ′ A � x ′ ⊗ 1 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, - dual A ∗ = A , - 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, - dual A ∗ = A , - biproduct A ⊕ B is the disjoint union, - ev A = { ( 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, - dual A ∗ = A , - biproduct A ⊕ B is the disjoint union, - ev A = { ( a ⊗ a, 0) | a ∈ A } , - coev A = { (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, - dual A ∗ = A , - biproduct A ⊕ B is the disjoint union, - ev A = { ( a ⊗ a, 0) | a ∈ A } , - coev A = { (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 coev A ⊗ µ 1 A ∗ ⊗ ∆ ⊗ 1 A ∗ A ∗ ⊗ A ⊗ A ∗ � A ∗ ⊗ A ⊗ A ⊗ A ∗ A ⊗ A τ ⊗ ev A ε ⊗ ε A ⊗ A ∗ 1 ev A 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 �→ [ δ gh : x �→ δ x,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 �→ [ δ gh : x �→ δ x,gh ] . Diagram chase for the augmented magma condition: coev A ⊗ µ 1 A ∗ ⊗ ∆ ⊗ 1 A ∗ ∑ � ∑ g ⊗ h ✤ x ∈ G δ x ⊗ x ⊗ δ gh x ∈ G δ x ⊗ x ⊗ x ⊗ δ gh ❴ ❴ ε ⊗ ε τ ⊗ ev A ∑ 1 = δ gh,gh x ∈ G δ x,gh ( x ⊗ δ x ) ev A

✤ � � ✤ � � 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 �→ [ δ gh : x �→ δ x,gh ] . Diagram chase for the augmented magma condition: coev A ⊗ µ 1 A ∗ ⊗ ∆ ⊗ 1 A ∗ ∑ � ∑ g ⊗ h ✤ x ∈ G δ x ⊗ x ⊗ δ gh x ∈ G δ x ⊗ x ⊗ x ⊗ δ gh ❴ ❴ ε ⊗ ε τ ⊗ ev A ∑ 1 = δ gh,gh x ∈ G δ x,gh ( x ⊗ δ x ) ev A Remark: If R = Z , then ε : g �→ 1 is the augmentation in Z G .

✤ � � ✤ � � 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 �→ [ δ gh : x �→ δ x,gh ] . Diagram chase for the augmented magma condition: coev A ⊗ µ 1 A ∗ ⊗ ∆ ⊗ 1 A ∗ ∑ � ∑ g ⊗ h ✤ x ∈ G δ x ⊗ x ⊗ δ gh x ∈ G δ x ⊗ x ⊗ x ⊗ δ gh ❴ ❴ ε ⊗ ε τ ⊗ ev A ∑ 1 = δ gh,gh x ∈ G δ x,gh ( x ⊗ δ x ) ev A Remark: If R = Z , then ε : g �→ 1 is the augmentation in Z G . In general, the augmentation need not be a counit for ∆.

Hypermagmas

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

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

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

Hypermagmas Consider set A with function A × A → 2 A ; ( 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 → 2 A ; ( 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 → 2 A ; ( 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 → 2 A ; ( 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 → 2 A ; ( 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 → 2 A ; ( 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 φ A,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 φ A,B,C : V ( B ⊗ A, C ) → V ( B, C ⊗ A ∗ ) at objects A, B, C of V . For an object A of V , define τ 13 : A 3 ⊗ A 2 ⊗ A 1 → A 1 ⊗ A 2 ⊗ A 3 ; a 3 ⊗ a 2 ⊗ a 1 �→ a 1 ⊗ a 2 ⊗ a 3

Currying and braiding in compact closed categories Compact closed category ( V , ⊗ , 1 ). Lemma: There is a natural isomorphism with components φ A,B,C : V ( B ⊗ A, C ) → V ( B, C ⊗ A ∗ ) at objects A, B, C of V . For an object A of V , define τ 13 : A 3 ⊗ A 2 ⊗ A 1 → A 1 ⊗ A 2 ⊗ A 3 ; a 3 ⊗ a 2 ⊗ a 1 �→ a 1 ⊗ a 2 ⊗ a 3 and τ 23 : A 1 ⊗ A 3 ⊗ A 2 → A 1 ⊗ A 2 ⊗ A 3 ; a 1 ⊗ a 3 ⊗ a 2 �→ a 1 ⊗ a 2 ⊗ a 3

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, 1 τ ∗ 13 φ A,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, 1 τ ∗ 13 φ A,A ⊗ A, 1 and left division ( structure ) λ : A ⊗ A → A ∗ with λ = µφ − 1 A,A ⊗ A, 1 τ ∗ 23 φ A,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, 1 τ ∗ 13 φ A,A ⊗ A, 1 and left division ( structure ) λ : A ⊗ A → A ∗ with λ = µφ − 1 A,A ⊗ A, 1 τ ∗ 23 φ A,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, 1 τ ∗ 13 φ A,A ⊗ A, 1 and left division ( structure ) λ : A ⊗ A → A ∗ with λ = µφ − 1 A,A ⊗ A, 1 τ ∗ 23 φ A,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 �→ [ δ xy : z �→ δ z,xy ].

(Quasi-)Group algebras as augmented quasigroups Group algebra RG had multiplication structure µ : RG ⊗ RG → RG ∗ ; x ⊗ y �→ [ δ xy : z �→ δ z,xy ]. µφ − 1 Thus RG,RG ⊗ RG,R : x ⊗ y ⊗ z �→ δ z,xy ,

(Quasi-)Group algebras as augmented quasigroups Group algebra RG had multiplication structure µ : RG ⊗ RG → RG ∗ ; x ⊗ y �→ [ δ xy : z �→ δ z,xy ]. µφ − 1 Thus RG,RG ⊗ RG,R : x ⊗ y ⊗ z �→ δ z,xy , µ φ − 1 RG,RG ⊗ RG,R τ ∗ whence 13 : z ⊗ y ⊗ x �→ δ z,xy = δ x,zy − 1 = δ x,z/y ,

(Quasi-)Group algebras as augmented quasigroups Group algebra RG had multiplication structure µ : RG ⊗ RG → RG ∗ ; x ⊗ y �→ [ δ xy : z �→ δ z,xy ]. µφ − 1 Thus RG,RG ⊗ RG,R : x ⊗ y ⊗ z �→ δ z,xy , µ φ − 1 RG,RG ⊗ RG,R τ ∗ whence 13 : z ⊗ y ⊗ x �→ δ z,xy = δ x,zy − 1 = δ x,z/y , so right division µ φ − 1 RG,RG ⊗ RG,R τ ∗ 13 φ RG,RG ⊗ RG,R = ρ : z ⊗ y �→ δ z/y .

(Quasi-)Group algebras as augmented quasigroups Group algebra RG had multiplication structure µ : RG ⊗ RG → RG ∗ ; x ⊗ y �→ [ δ xy : z �→ δ z,xy ]. µφ − 1 Thus RG,RG ⊗ RG,R : x ⊗ y ⊗ z �→ δ z,xy , µ φ − 1 RG,RG ⊗ RG,R τ ∗ whence 13 : z ⊗ y ⊗ x �→ δ z,xy = δ x,zy − 1 = δ x,z/y , so right division µ φ − 1 RG,RG ⊗ RG,R τ ∗ 13 φ RG,RG ⊗ RG,R = ρ : z ⊗ y �→ δ z/y . Similarly, have left division structure λ : x ⊗ z �→ δ x − 1 z = δ x \ z .

(Quasi-)Group algebras as augmented quasigroups Group algebra RG had multiplication structure µ : RG ⊗ RG → RG ∗ ; x ⊗ y �→ [ δ xy : z �→ δ z,xy ]. µφ − 1 Thus RG,RG ⊗ RG,R : x ⊗ y ⊗ z �→ δ z,xy , µ φ − 1 RG,RG ⊗ RG,R τ ∗ whence 13 : z ⊗ y ⊗ x �→ δ z,xy = δ x,zy − 1 = δ x,z/y , so right division µ φ − 1 RG,RG ⊗ RG,R τ ∗ 13 φ RG,RG ⊗ RG,R = ρ : z ⊗ y �→ δ z/y . Similarly, have left division structure λ : x ⊗ z �→ δ x − 1 z = δ x \ 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 �→ [ δ xy : z �→ δ z,xy ]. µφ − 1 Thus RG,RG ⊗ RG,R : x ⊗ y ⊗ z �→ δ z,xy , µ φ − 1 RG,RG ⊗ RG,R τ ∗ whence 13 : z ⊗ y ⊗ x �→ δ z,xy = δ x,zy − 1 = δ x,z/y , so right division µ φ − 1 RG,RG ⊗ RG,R τ ∗ 13 φ RG,RG ⊗ RG,R = ρ : z ⊗ y �→ δ z/y . Similarly, have left division structure λ : x ⊗ z �→ δ x − 1 z = δ x \ 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 �→ [ δ xy : z �→ δ z,xy ]. µφ − 1 Thus RG,RG ⊗ RG,R : x ⊗ y ⊗ z �→ δ z,xy , µ φ − 1 RG,RG ⊗ RG,R τ ∗ whence 13 : z ⊗ y ⊗ x �→ δ z,xy = δ x,zy − 1 = δ x,z/y , so right division µ φ − 1 RG,RG ⊗ RG,R τ ∗ 13 φ RG,RG ⊗ RG,R = ρ : z ⊗ y �→ δ z/y . Similarly, have left division structure λ : x ⊗ z �→ δ x − 1 z = δ x \ 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 y ≤ x → z , x ∧ y ≤ z ⇔ x ≤ y → 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 y ≤ x → z , x ∧ y ≤ z ⇔ x ≤ y → 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 y ≤ x → z , x ∧ y ≤ z ⇔ x ≤ y → 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 = N X in ( N , ⊗ , N ), augmented comagma ( A, ∆ , ε ) with diagonal ∆: x → x ⊗ x and augmentation ε : A → N .

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

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

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

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