A Study to Generate a Weak Order from a Partially Ordered Set, Taken. . .
79
• A catalogue could be aimed, where structures are gathered, which typically lead
to strong deviations
• As a candidate for a better approximation the method by Bubley and Dyer (1999),
may be selected and modified in that manner that the weak order as a result of
Eqs. 8 and 9 is a starting linear extension.
Summarizing, we hope that the present study has revealed some mathematical
ideas which may be of interest and attract new research by scholars of the
mathematical chemistry scene.
References
Annoni, P., Bruggemann, R., & Carlsen, L. (2014). A multidimensional view on poverty in the
European Union by partial order theory. Journal of Applied Statistics, 42, 535–554.
Atkinson, M. D., & Chang, H. W. (1986). Extensions of partial orders of bounded width.
Congressus Numerantium, 52, 21–35.
Bruggemann, R., & Carlsen, L. (2011). An improved estimation of averaged ranks of partially
orders. MATCH Communications in Mathematical and in Computer Chemistry, 65, 383–414.
Bruggemann, R., & Halfon, E. (1997). Comparative analysis of nearshore contaminated sites in
Lake Ontario: ranking for environmental hazard. Journal of Environmental Science and Health,
A32(1), 277–292.
Bruggemann, R., & Patil, G. P. (2011). Ranking and prioritization for multi-indicator systems –
Introduction to partial order applications. New York: Springer.
Bruggemann, R., & Voigt, K. (2008). Basic principles of hasse diagram technique in chemistry.
Combinatorial Chemistry & High Throughput Screening, 11, 756–769.
Bruggemann, R., & Voigt, K. (2011). A new tool to analyze partially ordered sets – Application:
Ranking of polychlorinated biphenyls and alkanes/alkenes in River main, Germany. MATCH
Communications in Mathematical and in Computer Chemistry, 66, 231–251.
Bruggemann, R., & Voigt, K. (2012). Antichains in partial order, example: Pollution in a German
region by lead, cadmium, zinc and sulfur in the herb layer. MATCH Communications in
Mathematical and in Computer Chemistry, 67, 731–744.
Bruggemann, R., Halfon, E., Welzl, G., Voigt, K., & Steinberg, C. (2001). Applying the concept of
partially ordered sets on the ranking of near-shore sediments by a battery of tests. The Journal
for Chemical Information and Computer scientists, 41, 918–925.
Bruggemann, R., Sørensen, P. B., Lerche, D., & Carlsen, L. (2004). Estimation of averaged ranks
by a local partial order model. The Journal for Chemical Information and Computer scientists,
44, 618–625.
Bruggemann, R., Voigt, K., Restrepo, G., & Simon, U. (2008). The concept of stability fields and
hot spots in ranking of environmental chemicals. Environmental Modelling & Software, 23,
1000–1012.
Bruggemann, R., Carlsen, L., Voigt, K., & Wieland, R. (2014). PyHasse software for partial order
analysis: Scientific background and description of selected modules. In R. Bruggemann, L.
Carlsen, & J. Wittmann (Eds.), Multi-indicator systems and modelling in partial order (pp.
389–423). New York: Springer.
Bubley, R., & Dyer, M. (1999). Faster random generation of linear extensions. Discrete Mathematics, 201, 81–88.
Buonocore, E., Mellino, S., De Angelis, G., Liu, G., & Ulgiati, S. (2018). Life cycle assessment
indicators of urban wastewater and sewage sludge treatment. Ecological Indicators, 94, 13–23.
Carlsen, L. (2008a). Hierarchical partial order ranking. Environmental Pollution, 155, 247–253.
79
• A catalogue could be aimed, where structures are gathered, which typically lead
to strong deviations
• As a candidate for a better approximation the method by Bubley and Dyer (1999),
may be selected and modified in that manner that the weak order as a result of
Eqs. 8 and 9 is a starting linear extension.
Summarizing, we hope that the present study has revealed some mathematical
ideas which may be of interest and attract new research by scholars of the
mathematical chemistry scene.
References
Annoni, P., Bruggemann, R., & Carlsen, L. (2014). A multidimensional view on poverty in the
European Union by partial order theory. Journal of Applied Statistics, 42, 535–554.
Atkinson, M. D., & Chang, H. W. (1986). Extensions of partial orders of bounded width.
Congressus Numerantium, 52, 21–35.
Bruggemann, R., & Carlsen, L. (2011). An improved estimation of averaged ranks of partially
orders. MATCH Communications in Mathematical and in Computer Chemistry, 65, 383–414.
Bruggemann, R., & Halfon, E. (1997). Comparative analysis of nearshore contaminated sites in
Lake Ontario: ranking for environmental hazard. Journal of Environmental Science and Health,
A32(1), 277–292.
Bruggemann, R., & Patil, G. P. (2011). Ranking and prioritization for multi-indicator systems –
Introduction to partial order applications. New York: Springer.
Bruggemann, R., & Voigt, K. (2008). Basic principles of hasse diagram technique in chemistry.
Combinatorial Chemistry & High Throughput Screening, 11, 756–769.
Bruggemann, R., & Voigt, K. (2011). A new tool to analyze partially ordered sets – Application:
Ranking of polychlorinated biphenyls and alkanes/alkenes in River main, Germany. MATCH
Communications in Mathematical and in Computer Chemistry, 66, 231–251.
Bruggemann, R., & Voigt, K. (2012). Antichains in partial order, example: Pollution in a German
region by lead, cadmium, zinc and sulfur in the herb layer. MATCH Communications in
Mathematical and in Computer Chemistry, 67, 731–744.
Bruggemann, R., Halfon, E., Welzl, G., Voigt, K., & Steinberg, C. (2001). Applying the concept of
partially ordered sets on the ranking of near-shore sediments by a battery of tests. The Journal
for Chemical Information and Computer scientists, 41, 918–925.
Bruggemann, R., Sørensen, P. B., Lerche, D., & Carlsen, L. (2004). Estimation of averaged ranks
by a local partial order model. The Journal for Chemical Information and Computer scientists,
44, 618–625.
Bruggemann, R., Voigt, K., Restrepo, G., & Simon, U. (2008). The concept of stability fields and
hot spots in ranking of environmental chemicals. Environmental Modelling & Software, 23,
1000–1012.
Bruggemann, R., Carlsen, L., Voigt, K., & Wieland, R. (2014). PyHasse software for partial order
analysis: Scientific background and description of selected modules. In R. Bruggemann, L.
Carlsen, & J. Wittmann (Eds.), Multi-indicator systems and modelling in partial order (pp.
389–423). New York: Springer.
Bubley, R., & Dyer, M. (1999). Faster random generation of linear extensions. Discrete Mathematics, 201, 81–88.
Buonocore, E., Mellino, S., De Angelis, G., Liu, G., & Ulgiati, S. (2018). Life cycle assessment
indicators of urban wastewater and sewage sludge treatment. Ecological Indicators, 94, 13–23.
Carlsen, L. (2008a). Hierarchical partial order ranking. Environmental Pollution, 155, 247–253.