About a definition of metric over an abelian linearly ordered group

A G-metric over an abelian linearly ordered group G = (G,⊙,≤) is a binary operation, d G , verifying suitable properties. We consider a particular G metric derived by the group operation ⊙ and the total weak order ≤, and show that it provides a base for the order topology associated to G

Reformulating Arrow’s Conditions in Terms of Cardinal Pairwise Comparison Matrices Defined Over a General Framework

In the paper, we deal with cardinal preferences of experts when these are expressed by means of Pairwise Comparison Matrices (PCMs). In order to obtain general results, suitable for several kinds of PCMs proposed in literature, we focus on PCMs defined over a general unifying framework, that is an Abelian linearly ordered group. In this framework, firstly, we aggregate several PCMs and we analyse how the aggregated PCM preserves some coherence levels, such as transitivity, weak consistency and consistency. Then, we reformulate Arrow’s conditions in terms of PCMs, and we provide two preference aggregation procedures for representing group preferences that give a social PCM and a social cardinal ranking, respectively. Finally, we analyse how these preference aggregation procedures satisfy reformulated Arrow’s conditions

Gingival Recessions and Periodontal Status after Minimum 2-Year-Retention Post-Non-Extraction Orthodontic Treatment

The objectives of this study were to assess gingival recessions (GR) and periodontal status in patients previously treated with non-extraction orthodontic treatment and retention at a follow-up of a minimum of two years after the end of treatment. Data from patients aged between 16 and 35 years with a previous non-extraction orthodontic treatment and at least 2 years of retention and full records before and after treatment were collected. The casts were digitalized using the 3Shape TRIOS® intraoral scanner and the Viewbox4 software was used for the measurements. The following parameters were scored: inclination of the lower and upper incisors (IMPA and I^SN) and anterior crowding (Little index). The included patients were recalled for a clinical periodontal follow-up examination and the following parameters were evaluated: buccal and lingual GR (mm) of incisors and canines, bleeding of probing score, plaque score, and gingival phenotype. The digital cast analysis showed a mean Little index of 7.78 (SD 5.83) and 1.39 (SD 0.79), respectively, before and after treatment. The initial and final cephalometric analyses showed an I^SN of 103.53° and 105.78° (SD 7.21) and IMPA of 91.3°and 95.1°, respectively. At the follow-up periodontal visits, the patients showed an overall low oral hygiene with bleeding at probing in 66.6% and plaque in the anterior area in 76.2% of patients. From the total examined 240 teeth of the frontal sextants, three patients had GR (from 1 to 6.5 mm): in the upper arch two at canines and one at central incisor, whereas in the lower arch two at central and one at lateral incisors. The gingival phenotype was thick in 55% of cases. The lingual-to-lingual retainers at follow-up were present in 61.9% of patients. A slight increased risk for buccal GR development was found only in correlation with the presence of fixed retainer and thin gingival phenotype mainly in patients with gingivitis. Thus, non-extraction orthodontic treatment performed with controlled forces and biomechanics seems to not affect the development of GR or the periodontal health after retention