We develop a direct method to recover an orthoalgebra from its poset of Boolean subalgebras. For this a new notion of direction is introduced. Directions are also used to characterize in purely order-theoretic terms those posets that are isomorphic to the poset of Boolean subalgebras of an orthoalgebra. These posets are characterized by simple conditions defining orthodomains and the additional requirement of having enough directions. Excepting pathologies involving maximal Boolean subalgebras of four elements, it is shown that there is an equivalence between the category of orthoalgebras and the category of orthodomains with enough directions with morphisms suitably defined. Furthermore, we develop a representation of orthodomains with enough directions, and hence of orthoalgebras, as certain hypergraphs. This hypergraph approach extends the technique of Greechie diagrams and resembles projective geometry. Using such hypergraphs, every orthomodular poset can be represented by a set of points and lines where each line contains exactly three points. 0 1 a a ′ Principal ideal subalgebras are of the form ↓ a ∪ ↑ a ′ for a ∈ B. They will play a central role throughout the paper. To describe their use, we begin with the order-theoretic characterization of ideal subalgebras given by Sachs [22].Lemma 2.5. [22, Theorem 1] The dual modular elements of Sub(B) are the ideal subalgebras. The least element ⊥ of the Boolean domain Sub(B) is {0, 1}, the largest element ⊤ is B, and the atoms of Sub(B) are the elements {0, a, a ′ , 1} for a = 0, 1. Hence there is a bijection between complementary pairs {a, a ′ } in B and elements of Sub(B) that are either ⊥ or an atom.Definition 2.6. Call an element of a poset with a least element basic if it is either an atom or the least element.Definition 2.7. For an element a of a Boolean algebra, we denote the Boolean subalgebraLater on, we shall use the same notation also when a is an element of an orthoalgebra.Lemma 2.8. For a Boolean algebra B, the basic elements of Sub(B) that are dual modular are x a where either a, a ′ is basic. In fact, they are principal ideal subalgebras.Proof. Follows immediately from Lemma 2.5.Our key definition is the following: Definition 2.9. For B a Boolean algebra, we define the mapping ϕ : B → (Sub(B)) 2 by ϕ(a) = (↓ a ∪ ↑ a ′ , ↓ a ′ ∪ ↑ a) .We call ϕ(a) the principal pair corresponding to a.
We describe a type system with mixed linear and non-linear recursive types called LNL-FPC (the linear/nonlinear fixpoint calculus). The type system supports linear typing which enhances the safety properties of programs, but also supports non-linear typing as well which makes the type system more convenient for programming. Just like in FPC, we show that LNL-FPC supports type-level recursion which in turn induces term-level recursion. We also provide sound and computationally adequate categorical models for LNL-FPC which describe the categorical structure of the substructural operations of Intuitionistic Linear Logic at all non-linear types, including the recursive ones. In order to do so, we describe a new technique for solving recursive domain equations within the category CPO by constructing the solutions over pre-embeddings. The type system also enjoys implicit weakening and contraction rules which we are able to model by identifying the canonical comonoid structure of all non-linear types. We also show that the requirements of our abstract model are reasonable by constructing a large class of concrete models that have found applications not only in classical functional programming, but also in emerging programming paradigms that incorporate linear types, such as quantum programming and circuit description programming languages.
We consider the functor C that to a unital C*-algebra A assigns the partial order set C(A) of its commutative C*-subalgebras ordered by inclusion. We investigate how some C*-algebraic properties translate under the action of C to order-theoretical properties. In particular, we show that A is finite dimensional if and only C(A) satisfies certain chain conditions. We eventually show that if A and B are C*-algebras such that A is finite dimensional and C(A) and C(B) are order isomorphic, then A and B must be *-isomorphic. underlying topological vector space, but with multiplication defined by (a, b) → ba, where (a, b) → ab denotes the original multiplication. Since C(A) is always isomorphic to C(A op ) as poset, for each C*-algebra A, the existence of Connes' C*-algebra Ac shows that the order structure of C(A) is not always enough in order to reconstruct A. More recent counterexamples can be found in [37] and [38].Nevertheless, there are still problems one could study. For instance, in [10] Döring and Harding consider a functor similar to C, namely the functor V assigning to a von Neumann algebra M the poset V(M ) of its commutative von Neumann subalgebras, and prove that one can reconstruct the Jordan structure, i.e., the anticommutator (a, b) → ab + ba, of M from V(M ). Similarly, in [17], it is shown that if C(A) and C(B) are order isomorphic, then there exists a quasi-linear Jordan isomorphism between Asa and Bsa, the sets of self-adjoint elements of A and B, respectively. Here quasi-linear means linear with respect to elements that commute. In [18], it is even shown that this quasi-linear Jordan isomorphism is linear when A and B are AW*-algebras.Moreover, one could replace C(A) by a structure with stronger properties. An example of such a structure is an active lattice, defined in [25], where Heunen and Reyes also show that this structure is strong enough to determine AW*-algebras completely.
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