Notes on orthoalgebras in categories John Harding and Taewon Yang - PowerPoint PPT Presentation

Notes on orthoalgebras in categories John Harding and Taewon Yang Department of Mathematical Sciences New Mexico State University BLAST 2013, Chapman University Aug. 8, 2013

Overview We show a certain interval in the (canonical) orthoalgebra D A of an object A in a category K arises from decompositions.

• What kind of category are we considering here? • How can we obtain the orthoalgebra of decompositions of an object in such a category?

� � � Categories K Consider a category K with finite products such that • I . projections are epimorphisms and • II .for any ternary product p q i : X 1 ˆ X 2 ˆ X 3 Ý Ñ X i q i Pt 1 , 2 , 3 u , the following diagram is a pushout in K : p q 2 , q 3 q X 1 ˆ X 2 ˆ X 3 X 2 ˆ X 3 r X 3 p q 1 , q 3 q � X 3 X 1 ˆ X 3 p X 3 where p X 3 and r X 3 are the second projections.

� � Decompositions • An isomorphism A Ý Ñ X 1 ˆ ¨ ¨ ¨ ˆ X n in K is called an n -ary decomposition of A . • For decompositions f : A Ý Ñ X 1 ˆ X 2 and g : A Ý Ñ Y 1 ˆ Y 2 of A , we say f is equivalent to g if there are isomorphisms γ i : X i Ý Ñ Y i p i “ 1 , 2 q such that the following diagram is commutative in K f � X 1 ˆ X 2 A γ 1 ˆ γ 2 id A � Y 1 ˆ Y 2 A g

Notation. Given A P K , rp f 1 , f 2 qs : equivalence class of f : A Ý Ñ X 1 ˆ X 2 . D p A q : all equivalence classes of all decompositions of A in K .

Partial operation ‘ on decompositions For rp f 1 , f 2 qs and rp g 1 , g 2 qs in D p A q , • rp f 1 , f 2 qs ‘ rp g 1 , g 2 qs is defined if there is a ternary decomposition p c 1 , c 2 , c 3 q : A Ý Ñ C 1 ˆ C 2 ˆ C 3 of A such that rp f 1 , f 2 qs “ rp c 1 , p c 2 , c 3 qqs and rp g 1 , g 2 qs “ rp c 2 , p c 1 , c 3 qqs . In this case, define the sum by rp f 1 , f 2 qs ‘ rp g 1 , g 2 qs “ rp c 1 , c 2 q , c 3 qs • Also, the equivalence classes rp τ A , id A qs and rp id A , τ A qs are distinguished elements 0 and 1 in D p A q , respectively, where τ A : A Ý Ñ T is the unique map into the terminal object T .

Orthoalgebras in K The following is due to Harding. • Proposition 1. The structure p D p A q , ‘ , 0 , 1 q is an orthoalgebra. An orthoalgebra is a partial algebra p A , ‘ , 0 , 1 q such that for all a , b , c P A , 1. a ‘ b “ b ‘ a 2. a ‘ p b ‘ c q “ p a ‘ b q ‘ c 3. For every a in A , there is a unique b such that a ‘ b “ 1 4. If a ‘ a is defined, then a “ 0 Note. BAlg Ĺ OML Ĺ OMP Ĺ OA

Intervals in D p A q For any decomposition p h 1 , h 2 q : A Ý Ñ H 1 ˆ H 2 of A in K , define the interval of p h 1 , h 2 q by L rp h 1 , h 2 qs “ trp f 1 , f 2 qs P D p A q | rp f 1 , f 2 qs ď rp h 1 , h 2 qsu , where ď is the induced order from the orthoalgebra D p A q , that is, • rp f 1 , f 2 qs ď rp h 1 , h 2 qs means rp f 1 , f 2 qs ‘ rp g 1 , g 2 qs “ rp h 1 , h 2 qs for some decomposition p g 1 , g 2 q of A in K .

Intervals as decompositions Proposition 2.(HY) For each decomposition p h 1 , h 2 q : A Ý Ñ H 1 ˆ H 2 of an object A in K , the interval L rp h 1 , h 2 qs is isomorphic to D p H 1 q .

Example 1 • The category Grp of all groups and their maps satisfies all the necessary hypothesis. Consider a cyclic group G “ x a y of order 30. Notice | G | “ 2 ¨ 3 ¨ 5. x a y x a 2 y x a 3 y x a 5 y x a 6 y x a 10 y x a 15 y t e u

• D p G q in the category Grp . px a y , x e yq px a 2 y , x a 15 yq px a 3 y , x a 10 yq px a 5 y , x a 6 yq px a 6 y , x a 5 yq px a 10 y , x a 3 yq px a 15 y , x a 3 yq px e y , x a yq

• The interval L px a 2 y , x a 15 yq is a four element Boolean lattice. Also, we have the following: D px a 2 yq – tpx a 6 y , x a 10 yq , px a 10 y , x a 6 yq , px a 2 y , x e yq , px e y , x a 2 yqu Thus we obtain L px a 2 y , x a 15 yq – D px a 2 yq

Example 2 • Consider the cyclic group G “ x a y with | G | “ 12 “ 4 ¨ 3. x a y x a 2 y x a 3 y x a 4 y x a 6 y x e y

Factor pairs : tpx a 4 y , x a 3 yq , px a 3 y , x a 4 yqpx e y , x a yq , px a y , x e yqu (four-element Boolean lattice. Note that the poset is not isomorphic to Sub p G q ) L px a 3 y , x a 4 yq – 2 and D px a 3 yq – 2

� Proof (Sketch) The essential part of the proof is to construct maps F and G F � D p H 1 q L rp h 1 , h 2 qs G

First define G : D p H 1 q Ý Ñ L rp h 1 , h 2 qs by rp m 1 , m 2 qs ù rp m 1 h 1 , p m 2 h 1 , h 2 qqs

Conversely, seeking a map F : L rp h 1 , h 2 qs Ý Ñ D p H 1 q , consider a binary decomposition p f 1 , f 2 q : A Ý Ñ F 1 ˆ F 2 in L rp h 1 , h 2 qs . Then there is an isomorphism p c 1 , c 2 , c 3 q : A Ý Ñ C 1 ˆ C 2 ˆ C 3 in K such that rp f 1 , f 2 qs “ rp c 1 , p c 2 , c 3 qqs and rp h 1 , h 2 qs “ rpp c 1 , c 2 q , c 3 qs The latter implies that there is an isomorphism p r 1 , r 2 q : H 1 Ý Ñ C 1 ˆ C 2 with p r 1 , r 2 q h 1 “ p c 1 , c 2 q . Then define the map F by rp f 1 , f 2 qs ù rp r 1 , r 2 qs

It is known-that the correspondences F and G are indeed well-defined. Moreover, they are orthoalgebra homomorphisms that are inverses to each other.

Speculations • Do we have more instances for the conditions I and II? • Can we give some categorical conditions on morphisms so that D p A q is an orthomodular poset? Moreover, can we also give some order/category-theoretic conditions on Sub p A q in K such that Sub p A q Ý Ñ D p A q is an orthomodular embedding (For example, Hilb K -like category)?

References • M. Dalla Chiara, R. Giuntini, and R. Greechie, Reasoning in quantum theory, Sharp and unsharp quantum logics , Trends in Logic-Studia Logica Library, 22. Kluwer Academic Publishers, Dordrecht, 2004. • F. W. Lawvere and R. Rosebrugh, Sets for mathematics , Cambridge University Press, Cambridge, 2003. • J. Flachsmeyer, Note on orthocomplemented posets , Proceedings of the Conference, Topology and Measure III. Greifswald, Part 1, 65-73, 1982. • J. Harding, Decompositions in quantum logic ,Trans. Amer. Math. Soc. 348 (1996), no. 5, 1839-1862. • T. Yang, Orthoalgebras , Manuscript, 2009.

Thank you