Injective Hulls of Semilattices

Bruns, Günter; Lakser, H.

doi:10.4153/cmb-1970-023-6

Cited by 75 publications

(47 citation statements)

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“…In case the property lattice would be a complete Heyting algebra in which all joins encode disjunctions, then A and { A} again mean the same thing. As argued in Coecke (2001a), this redundancy is then exactly eliminated by considering distributive ideals DI(L) (Bruns and Lakser 1970), that is, order ideals, that are closed under joins of distributive sets (abbreviated as distributive joins), i.e. if A ⊆ I ∈ DI(L) then A ∈ I whenever we have ∀b ∈ L : b ∧ A = {b ∧ a | a ∈ A}.…”

Section: True Implicative Quantum Logicalitymentioning

confidence: 99%

“…state, possesses a quality a if it is the case that: whenever it (in realization p) is within environment e a then it causes phenomenon α a to happen" and it is by this statement that we identify a particular quality of the system 10 -this explicit consideration of the environment (or context), even in the system's endo-perspective, is what gives the operational flavour to this approach. 11 Before we continue let us first recall some basic order theoretical notions. A complete lattice is a bounded partially ordered set (L, ≤, 0, 1) which is such that every subset A ⊆ L has a greatest lower bound or meet A.…”

Section: What Quantum Logicality Can Be Aboutmentioning

confidence: 99%

“…Again, by the passive formulation we avoid any connotation with some role that is in many interpretations of quantum theory ascribed to the so-called 'observer'. 11 Note that operationalism has here nothing to do with instrumentalism. In Piron's formulation the tendency towards an instrumentalist interpretation is however a bit stronger due to the explicit presence of 'definite experimental projects'.…”

Section: What Quantum Logicality Can Be Aboutmentioning

confidence: 99%

“…Note here that for the particular example of quantum theory the condition is trivially satisfied as it is the case for any atomistic property lattice. 24 Now, any complete lattice L has DI(L) of distributive ideals as its distributive hull (Bruns and Lakser 1970), providing the construction with a (quasi-)universal property. 25 Moreover, DI(L) itself always proves to be a complete Heyting algebra and the inclusion preserves all meets and existing distributive joins.…”

Section: Proposition 2 If "Existence Of Superposition States Impliesmentioning

confidence: 99%

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The Sasaki Hook Is Not a [Static] Implicative Connective but Induces a Backward [in Time] Dynamic One That Assigns Causes

Coecke

Smets²

2004

International Journal of Theoretical Physics

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In this paper we argue that the Sasaki adjunction, which formally encodes the logicality that different authors tried to attach to the Sasaki hook as a 'quantum implicative connective', has a fundamental dynamic nature and encodes the so-called 'causal duality' (Coecke, Moore and Stubbe 2001) for the particular case of a quantum measurement with a projector as corresponding self-adjoint operator. In particular: The action of the Sasaki hook (a S → −) for fixed antecedent a assigns to some property "the weakest cause before the measurement of actuality of that property after the measurement", i.e. (a S → b) is the weakest property that guarantees actuality of b after performing the measurement represented by the projector that has the 'subspace a' as eigenstates for eigenvalue 1 , say, the measurement that 'tests' a . From this we conclude that the logicality attributable to quantum systems contains a fundamentally dynamic ingredient: Causal duality actually provides a new dynamic interpretation of orthomodularity. We also reconsider the status of the Sasaki hook within 'dynamic (operational) quantum logic' (DOQL), what leads us to the claim made in the title of this paper. More explicitly, although (as many argued in the past) the Sasaki hook should not be seen as an implicative hook, the formal motivation that persuaded others to do so, i.e. the Sasaki adjunction, does have a physical significance (in terms of causal duality). It is within the context of DOQL that we can then derive that the labeled dynamic hooks (forwardly and backwardly) that encode quantum measurements act on properties as (a 1, taking values in the 'disjunctive extension' DI(L) of the property lattice L (Coecke 2001a) , where a ∈ L is the tested property and (− → L −) is the Heyting implication that lives on DI(L) . Since these hooks (− ϕa → −) and (− ϕa ← −) extend to DI(L) × DI(L) they constitute internal operations . In an even more radical perspective one could say that the transition from either classical or constructive/intuitionistic logic to quantum logic entails besides the introduction of an additional unary connective 'operational resolution' (Coecke 2001a) the shift from a binary connective implication to a ternary connective where two of the arguments refer to qualities of the system and the third, the new one, to an obtained outcome (in a measurement). QUANTUM LOGICALITYWe claim that logical considerations on quantum behavior and as such, further development of the research field, have been 'corrupted' by two features. Once these two features are neutralized, the way towards an essentially dynamic quantum logic (i.e. a unified logic of 'changes' both for classical and quantum systems), or otherwise put, a true quantum process semantics, is opened. Moreover, the solution to the second 'corrupt feature' indicates that the logicality encoded in pure quantum theory is of a fundamental dynamic nature. Structures somewhat similar to those emerging in the context of categorical grammar (Lambek 1958), linear logic (Girard 1987...

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Section: True Implicative Quantum Logicalitymentioning

confidence: 99%

Section: What Quantum Logicality Can Be Aboutmentioning

confidence: 99%

Section: What Quantum Logicality Can Be Aboutmentioning

confidence: 99%

Section: Proposition 2 If "Existence Of Superposition States Impliesmentioning

confidence: 99%

See 2 more Smart Citations

The Sasaki Hook Is Not a [Static] Implicative Connective but Induces a Backward [in Time] Dynamic One That Assigns Causes

Coecke

Smets²

2004

International Journal of Theoretical Physics

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“…Using the characterization [3], [18] of injective A-semilattices S as complete Brouwerian lattices and the fact that for any A E S, I0(A) is a complete join-continuous lattice (see [10, p. 152]), we quickly see that 7004) is an injective A-semilattice iff it is a distributive lattice. Therefore A is flat iff I0(A) is a distributive lattice.…”

mentioning

confidence: 99%

Flat Semilattices

Bulman-Fleming¹,

Dowell²

1978

Proceedings of the American Mathematical Society

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Abstract. Let S (respectively So) denote the category of all join-semilattices (resp. join-semilattices with 0) with (0-preserving) semilattice homomorphisms. For A G S let A0 represent the object of S0 obtained by adjoining a new 0-element. In either category the tensor product of two objects may be constructed in such a manner that the tensor product functor is left adjoint to the hom functor. An object A eS (Sq) is called flat if the functor -®gA (-OE^) preserves monomorphisms in S (So).Theorem. For A 6 S (S0) the following conditions are equivalent: (1) A is fiat in S (Srj), (2) A0 (A) is distributive {see Grätzer, Lattice theory,p. 117), (3) A is a directed colimit of a system of f.g. free algebras in S (So). The equivalence of (1) and (2) in S was previously known to James A. Anderson.(1) «=> (3) is an analogue of Lazard's well-known result for Ä-modules.

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