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Gerardo Beltran Serrano,1– 3 Laura Pooch Rodrigues,2 Bruno Schein,1,2 Maxciel Zortea,1,2 Iraci Lucenada Silva Torres,1,4,5 Felipe Fregni,6 Wolnei Caumo1,2,5 1Post-Graduate Program in Medical Sciences, School of Medicine, Universidade Federal Do Rio Grande Do Sul (UFRGS), Porto Alegre, Brazil; 2Laboratory of Pain and Neuromodulation at Hospital De Clínicas De Porto Alegre (HCPA), Porto Alegre, Brazil; 3Psychology Department, Universidad Catolica De Cuenca, UCACUE, Cuenca, Ecuador; 4Department of Pharmacology, Institute of Health Sciences (ICBS), Universidade Federal Do Rio Grande Do Sul (UFRGS), Porto Alegre, Brazil; 5Pharmacology of Pain and Neuromodulation: Pre-Clinical Investigations Research Group, Universidade Federal Do Rio Grande Do Sul (UFRGS), Porto Alegre, Brazil; 6Department of Neurology, Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, USACorrespondence: Wolnei Caumo Fax + (55) 51- 3359.8083Email wcaumo@hcpa.edu.brObjective: We evaluated whether active(a)-tDCS combined with hypnotic analgesia suggestion (HS) would be more effective than a single active(a)-tDCS, and/or sham-(s)-tDCS and s-tDCS/HS on the following outcomes: function of descending pain modulatory system (DPMS) during the conditioned pain modulation test (CPM-test) (primary outcome), heat pain threshold (HPT), heat pain tolerance (HPTo) and cold pressor test (CPT) (secondary outcomes). We also examined whether their effects are related to neuroplasticity state evaluated by serum brain-derived-neurotropic factor (BDNF).Materials and Methods: Forty-eight females received one session of one of the four interventions (a-tDCS/HS, s-tDCS/HS, a-tDCS, and s-tDCS) in an incomplete randomized crossover sequence. The a-tDCS or s-tDCS was applied over the left dorsolateral prefrontal cortex (DLPFC) for 30 minutes at 2mA.Results: A generalized linear model revealed a significant main effect for the intervention group (P < 0.032). The delta-(Δ) pain score on the Numerical Pain Scale (NPS0-10) during CPM-test in the a-tDCS/HS group was − 0.25 (0.43). The (Δ) pain score on NPS (0– 10) during CPM-test in the other three groups was a-tDCS=− 0.54 (0.41), HS − 0.01 (0.41) and s-tDCS/HS=− 0.19 (0.43). A-tDCS/HS intervention increased the CPT substantially compared to all other interventions. Also, higher baseline levels of BDNF were associated with a larger change in CPT and HPTo.Conclusion: These findings indicate that the HS combined with a-tDCS mitigated the effect of the a-tDCS on the DPMS. The a-tDCS up-regulates the inhibition on DPMS, and the HS improved pain tolerance. And, together they enhanced the reaction time substantially upon the CPT.Clinical Trial Registration: www.ClinicalTrials.gov, identifier NCT03744897.Keywords: tDCS, hypnotic analgesia, conditioned pain modulation, pain perception

In his seminal article of 1941, Paul Dubreil introduced \textit{complexes forts} of semigroups. Strong subsets of a semigroup $S$ form another semigroup under a natural multiplication. Properties of this semigroup are studied and some open problems raised (specially when $S$ is a group or an inverse semigroup). Also, a simple proof of a known result is given: every inverse semigroup can be isomorphically embedded in the semigroup of cosets of a group.