Background

Wehrung, Friedrich

doi:10.1007/978-3-319-61599-8_1

Cited by 3 publications

(2 citation statements)

References 18 publications

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“…A homomorphism of Boolean inverse semigroups is called additive if it preserves joins of orthogonal idempotents, in which case it preserves all finite joins existing in S . There are a number of other axiomatizations of Boolean inverse semigroups, and we refer the reader to [43] for more details.…”

Section: Groupoids and Inverse Semigroupsmentioning

confidence: 99%

Twists, Crossed Products and Inverse Semigroup Cohomology

Steinberg

2021

J. Aust. Math. Soc.

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Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists. We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

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Section: Groupoids and Inverse Semigroupsmentioning

confidence: 99%

Twists, Crossed Products and Inverse Semigroup Cohomology

Steinberg

2021

J. Aust. Math. Soc.

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“…The following supporting information can be downloaded at: , Figure S1: Set/Subset Relationships; Table S1: A list of lower sets and singletons; Table S2: Lower-set Hierarchy; Table S3: The Inversions; Table S4: All possible “lower-bound”. References [ 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 ] are cited in the supplementary materials.…”

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confidence: 99%

The Structure and Organizations of ICHD-3 Differential Diagnoses through DiffNet: A Pilot Study

Zhang

2022

Diagnostics

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Differential diagnosis is fundamental to medicine. Using DiffNet, a differential diagnosis generator, as a model, we studied the structure and organization of how collections of diagnoses (i.e., sets of diagnoses) are related in the ICHD3. Furthermore, we explored the clinical and theoretical implication of these answers. Methods: DiffNet is a freely distributed differential diagnosis generator for headaches using graph theoretical properties of ICHD3: (1) DiffNet considers each ICHD3 diagnosis as a node. (2) An edge exists between two ICHD3 diagnoses if they are connected by either classification hierarchy or are cross-referenced in ICHD3 comment section. In the current project, we generated a set of differential diagnoses using DiffNet for each ICHD3 diagnosis. We then determined algorithmically the set/subset relationship between these sets. We also determined the smallest list of ICHD3 diagnosis whose differential diagnoses would encompass the totality of ICHD3 diagnoses. Results: All ICHD3 diagnoses can be represented by a minimum of 92 differential diagnosis sets. Differential diagnosis sets for 10 of the 14 first digit subcategories of ICHD3 are represented by more than one differential diagnosis sets. Fifty-one of the 93 differential diagnosis sets contain multiple subset relationships; the remaining 42 do not enter into any set/subset relationship with other differential diagnosis sets. Finally, we included a hierarchical presentation of differential diagnosis sets in ICHD3 according to DiffNet. Conclusion: We propose a way of interpreting headache differential diagnoses as partial ordered sets (i.e., poset). For clinicians, fluency in the 93 diagnoses and their differential, as put forth here, implies a complete description of ICHD3. On a theoretical level, interpreting ICHD3 differential diagnosis as poset allows for researchers to translate differential diagnoses sets topologically, algebraically, and categorically.

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The structure and organization of headache differential diagnoses: A Pilot Study of Subset Relationships between Differentials in ICHD3

Zhang

2021

Preprint

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Structured AbstractObjectiveDifferential diagnosis is fundamental to medicine. Using DiffNet, a differential diagnosis generator, as a model we studied the structure and organization of how collections of diagnose (i.e. sets of diagnoses) are related in the ICHD3. Specifically, we asked: Which sets of differential diagnoses are subsets of each other? What is the minimum number of sets of differential diagnoses that encompass all ICHD3 codes? Furthermore, we explored the clinical and theoretical implication of these answers.MethodsDiffNet is a freely distributed differential diagnosis generator for headaches using graph theoretical properties of ICHD3. For each ICHD3 diagnosis, we generated a set of differential diagnoses using DiffNet. We then determined algorithmically the set/subset relationship between these sets. We also determined the smallest list of ICHD3 diagnosis whose differential diagnoses would encompass the totality of ICHD3 diagnoses.ResultsAll ICHD3 diagnoses can be represented by a minimum of 92 differential diagnosis sets. Differential diagnosis sets for 10 of the 14 first digit subcategories of ICHD3 are represented by more than one differential diagnosis sets. Fifty-one of the 93 differential diagnosis sets contain multiple subset relationships; the remaining 42 do not enter into any set/subset relationship with other differential diagnosis sets. Finally, we included a hierarchical presentation of differential diagnosis sets in ICHD3 according to DiffNet.ConclusionWe propose a way of interpreting headache differential diagnoses as partial ordered sets (i.e. poset). For clinicians, fluency with the 93 diagnoses and their differential put forth here implies a complete description of ICHD3. On a theoretical level, interpreting ICHD3 differential diagnosis as poset, allows researchers to translate differential diagnoses sets topologically, algebraically, and categorically.

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Background

Cited by 3 publications

References 18 publications

Twists, Crossed Products and Inverse Semigroup Cohomology

Twists, Crossed Products and Inverse Semigroup Cohomology

The Structure and Organizations of ICHD-3 Differential Diagnoses through DiffNet: A Pilot Study

The structure and organization of headache differential diagnoses: A Pilot Study of Subset Relationships between Differentials in ICHD3

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