The lattices of Moore families and closure operators on a finite set: A survey. (English) Zbl 0971.06009

Caspard, Nathalie; Monjardet, Bernard

Janowitz, Melvin F. (ed.), OSDA98. Ordinal and symbolic data analysis. Univ. of Massachusetts, Amherst, MA, USA, September 28-30, 1998. Amsterdam: Elsevier, Electron. Notes Discrete Math. 2, no pag., electronic only (1999).

Summary: We present a survey of properties of the lattice of Moore families (families of subsets of a set \(S\) containing \(S\) and closed by set intersection) on a finite set \(S\), with proofs of the more significant results. In particular, we prove that this lattice is atomistic and lower bounded and that there exists a canonical basis allowing to represent any Moore family by “implicational” Moore families. The notion of Moore family has many cryptomorphic versions, especially the notions of closure operator and of (full) implicational system, which occur in many fields in pure or applied mathematics and computer science.
For the entire collection see [Zbl 0968.00024].

MSC:

06B05	Structure theory of lattices
06A15	Galois correspondences, closure operators (in relation to ordered sets)

Keywords:

lattice of Moore families; closure operator; implicational system

Software:

SAS; SAS/STAT; C4.5

References:

[1]	Blind (Afb), American Foundation For The: Directory of services for the blind and visually impaired persons in the united states and Canada. (1993)
[2]	Bay, K. S.; Leatt, P.; Stinson, S. M.: A patient-classification system for long-term care. Medical care 20, No. 5, 468-488 (1982)
[3]	Berlowitz, D. R.; Rosen, A. K.; Moskowitz, M. A.: Ambulatory care casemix measures. Journal of general internal medicine 10, No. 3, 162-170 (1995)
[4]	Bock, H. H.: Classification and related methods of data analysis. (1988) · Zbl 0732.00020
[5]	Bratko, I.; Mozetic, I.; Lavrac, L.: KARDIO: A study in deep and qualitative knowledge for expert systems. (1989)
[6]	Breiman, L.; Friedman, J. H.; Olshen, J. H.; Stone, C. J.: Classification and regression trees. (1984) · Zbl 0541.62042
[7]	Cameroun, J. M.; Baraff, L. J.; Sekhon, R.: Case-mix classification for emergency departments.”. Medical care 28, 146-158 (1990)
[8]	Davis, M. M.: How long should A customer wait for service?. Decision sciences 22, 421-434 (1991)
[9]	Dilts, D. M.; Khamalah, J. N.; Plotkin, A. D.: Clinical low vision resource usage prediction. Optometry and vision science 71, No. 7, 422-436 (1994)
[10]	Dilts, D. M.; Khamalah, J. N.; Plotkin, A. D.: Using cluster analysis for medical decision making. Medical decision making 15, No. 4, 333-347 (1995)
[11]	Fetter, R. B.; Youngsoo, S.; Freeman, J. L.; Averill, R. F.; Thompson, J. D.: Case mix definition by diagnosis-related groups. Medical care 18, No. 2 Supplement, 1-53 (1980)
[12]	Fetter, R. B.; Freeman, J. L.: Diagnosis-related groups: product line management within hospitals. Academy of management review 11, No. 1, 41-54 (1986)
[13]	Fetter, R. B.: Diagnosis-related groups: understanding hospital performance. Interfaces 21, No. 1, 6-26 (1991)
[14]	Fetter, R. B.; Brand, D. A.; Gamache, D. E.: Drgs: their design and development. (1991)
[15]	Fitzsimmons, J. A.; Fitzsimmons, M. J.: Service management for competitive advantage. (1994) · Zbl 0979.26007
[16]	Fliedner, E. B.; Mabert, V. A.: Constrained forecasting: some implementation guidelines. Decision sciences 23, 1143-1161 (1992)
[17]	Gold, M.: Common sense on extending DRG concepts to pay for ambulatory care. Inquiry 25, No. 2, 281-289 (1988)
[18]	Gordon, A. D.: Classification. (1981) · Zbl 0507.62057
[19]	Hair, J. F.; Anderson, R. E.; Tathan, R. L.; Black, W. C.: Multivariate data analysis: with readings. (1995)
[20]	Hornbrook, M. C.; Hurtado, A.; Johnson, R. E.: Health care episodes: definition, measurement and use. Medical care review 42, 163-218 (1985)
[21]	Khamalah, J. and Dilts, D. M., ”Predicting Patient Resource Needs: A Methodology for Ambulatory Care”, Proceedings of the Decision Sciences National Meeting, San Diego, CA, (1997) 1464–1466.
[22]	Kolb, D.; Clay, S. B.: Ambulatory care groupings: when, how, and the impact on managed care. Journal of ambulatory care management 17, No. 1, 29-38 (1994)
[23]	Li, E. Y.: Artificial neural networks and their business applications,”. Information and management 27, No. 5, 303-359 (1994)
[24]	Macstravic, R. S.: Forecasting health care services: A provider’s guide. (1984)
[25]	Masters, T.: Neural network recipes in C++. (1993) · Zbl 0818.68049
[26]	Michalski, R. S.; Chilauski, R. L.: Knowledge acquisition by encoding expert rules versus computer induction from examples: A case study involving soybean pathology. International journal of man-machine studies 12, 63-87 (1980)
[27]	Michie, D.; Spiegelhalter, D. J.; Taylor, C. C.: Machine learning, neural and statistical classification. (1994) · Zbl 0827.68094
[28]	Quinlan, J. R.: Simplifying decision trees. International journal of man-machine learning 27, 221-234 (1987)
[29]	Quinlan, J. R.: C4.5: programs for machine learning. (1993)
[30]	Sas: SAS/STAT user’s guide, version 6. (1990)
[31]	Shapiro, A. D.; Michie, D.: A self-commenting facility for inductively synthesized end-game expertise. Advances in computer chess 5 (1986)
[32]	Starfield, B.; Weiner, J.; Mumford, L.; Steinwachs, D.: Ambulatory care groups: A categorization of diagnoses for research and management. Health services research 26, No. 1, 53-74 (1991)
[33]	Strong, J. G.; Pace, R. J.; Plotkin, A. D.: Low vision services: A model for sequential and rehabilitation. Canadian journal of public health 79, 50-54 (1988)
[34]	Tenan, P. M.; Fillmore, H. H.; Caress, B.; Kelly, W. P.; Nelson, H.: Pacs: classifying ambulatory care patients and services for clinical and financial management. Journal of ambulatory care management 11, No. 3, 36-53 (1988)
[35]	Tierney, W. M.; Fitzgerald, J. F.; Miller, E. M.; James, M. K.; Mcdonald, C. J.: Predicting inpatient costs with admitting clinical data. Medical care 33, No. 1, 1-14 (1995)
[36]	Weiss, S. M.; Kulikowski, C. A.: Computer systems that learn: classification and prediction methods from statistics, neural nets, machine learning, and expert systems. (1991)
[37]	Bank, World: World bank development report 1993: investing in health - world development indicators. (1993)