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A representation theorem for integral rigs and its applications to residuated lattices J.L. Castiglioni M. Menni W. J. Zuluaga Botero Universidad Nacional de La Plata CONICET SYSMICS Barcelona, September 2016 SYSMICS Barcelona, September

  1. A representation theorem for integral rigs and its applications to residuated lattices J.L. Castiglioni M. Menni W. J. Zuluaga Botero Universidad Nacional de La Plata CONICET SYSMICS Barcelona, September 2016 SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  2. Rigs and really local rigs Definition A rig is a structure ( A , · , 1 , + , 0 ) such that ( A , · , 1 ) and ( A , + , 0 ) are commutative monoids and distributivity holds in the sense that a · 0 = 0 and ( a + b ) · c = a · c + b · c for all a , b , c ∈ A . SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  3. � � Rigs and really local rigs Definition A rig is a structure ( A , · , 1 , + , 0 ) such that ( A , · , 1 ) and ( A , + , 0 ) are commutative monoids and distributivity holds in the sense that a · 0 = 0 and ( a + b ) · c = a · c + b · c for all a , b , c ∈ A . Let E be a category with finite limits. For any rig A in E we define the subobject Inv ( A ) → A × A by declaring that the diagram below ! � 1 Inv ( A ) 1 � A A × A · is a pullback. SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  4. � � Rigs and really local rigs Definition A rig is a structure ( A , · , 1 , + , 0 ) such that ( A , · , 1 ) and ( A , + , 0 ) are commutative monoids and distributivity holds in the sense that a · 0 = 0 and ( a + b ) · c = a · c + b · c for all a , b , c ∈ A . Let E be a category with finite limits. For any rig A in E we define the subobject Inv ( A ) → A × A by declaring that the diagram below ! � 1 Inv ( A ) 1 � A A × A · is a pullback. The two projections Inv ( A ) → A are mono in E and induce the same subobject of A . SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  5. � � Rigs and really local rigs Definition A rig morphism f : A → B between rigs in E is local if the following diagram � Inv B Inv A � B A f is a pullback. SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  6. � � � � Rigs and really local rigs Definition A rig morphism f : A → B between rigs in E is local if the following diagram � Inv B Inv A � B A f is a pullback. If E is a topos with subobject classifier ⊤ : 1 → Ω then there exists a unique map ι : A → Ω such that the square below ! � 1 Inv ( A ) ⊤ � Ω A ι is a pullback. SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  7. Rigs and really local rigs Definition (Lawvere, [1]) The rig A in E is really local if ι : A → Ω is local. SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  8. Rigs and really local rigs Definition (Lawvere, [1]) The rig A in E is really local if ι : A → Ω is local. An application of the internal logic of toposes shows the following: SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  9. Rigs and really local rigs Definition (Lawvere, [1]) The rig A in E is really local if ι : A → Ω is local. An application of the internal logic of toposes shows the following: Lemma The rig A is really local if and only if the following sequents hold 0 ∈ Inv ( A ) ⊢ ⊥ ( x + y ) ∈ Inv ( A ) ⊢ x , y x ∈ Inv ( A ) ∨ y ∈ Inv ( A ) x ∈ Inv ( A ) ∨ y ∈ Inv ( A ) ⊢ x , y ( x + y ) ∈ Inv ( A ) in the internal logic of E . SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  10. Integral rigs and really local integral rigs Definition A rig is called integral if the equation 1 + x = 1 holds. SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  11. Integral rigs and really local integral rigs Definition A rig is called integral if the equation 1 + x = 1 holds. SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  12. Integral rigs and really local integral rigs Definition A rig is called integral if the equation 1 + x = 1 holds. In every integral rig A the relation a ≤ b if and only if a + b = b , determines a partial order. SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  13. Integral rigs and really local integral rigs Definition A rig is called integral if the equation 1 + x = 1 holds. In every integral rig A the relation a ≤ b if and only if a + b = b , determines a partial order. Moreover, respect to this order ( A , + , 0 ) becomes a join-semilattice. SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  14. Integral rigs and really local integral rigs Definition A rig is called integral if the equation 1 + x = 1 holds. In every integral rig A the relation a ≤ b if and only if a + b = b , determines a partial order. Moreover, respect to this order ( A , + , 0 ) becomes a join-semilattice. Lemma If A is integral then the canonical 1 → Inv ( A ) is an iso. SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  15. Integral rigs and really local integral rigs Definition A rig is called integral if the equation 1 + x = 1 holds. In every integral rig A the relation a ≤ b if and only if a + b = b , determines a partial order. Moreover, respect to this order ( A , + , 0 ) becomes a join-semilattice. Lemma If A is integral then the canonical 1 → Inv ( A ) is an iso. Lemma (Really local integral rigs) An integral rig is really local if and only if the following sequents hold 0 = 1 ⊢ ⊥ x + y = 1 ⊢ x , y x = 1 ∨ y = 1 in the internal logic of E . SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  16. Integral rigs in Shv ( D ) Let D a bounded distributive lattice and Shv ( D ) the category of sheaves over D with the coherent topology. SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  17. Integral rigs in Shv ( D ) Let D a bounded distributive lattice and Shv ( D ) the category of sheaves over D with the coherent topology. In Shv ( D ) , an integral rig is a functor SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  18. Integral rigs in Shv ( D ) Let D a bounded distributive lattice and Shv ( D ) the category of sheaves over D with the coherent topology. In Shv ( D ) , an integral rig is a functor F : D op − → iRig SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  19. Integral rigs in Shv ( D ) Let D a bounded distributive lattice and Shv ( D ) the category of sheaves over D with the coherent topology. In Shv ( D ) , an integral rig is a functor F : D op − → iRig such that the composition with the forgetful functor iRig − → Set SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

  20. Integral rigs in Shv ( D ) Let D a bounded distributive lattice and Shv ( D ) the category of sheaves over D with the coherent topology. In Shv ( D ) , an integral rig is a functor F : D op − → iRig such that the composition with the forgetful functor iRig − → Set is a sheaf respect to the coherent topology. SYSMICS Barcelona, September 2016 J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero (UNLP) A representation theorem for integral rigs and its applications to residuated lattices / 24

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