Semigrupos numéricos

Um semigrupo numérico é um submonoide de (N, +) tal que o seu complementar em N é finito. Neste trabalho estudamos alguns invariantes de um semigrupo numérico S tais como: multiplicidade, dimensão de imersão, número de Frobenius, falhas e conjunto Apéry de S. Caracterizamos uma apresentação minimal para um semigrupo numérico S e descrevemos um método algorítmico para determinar esta apresentação. Definimos um semigrupo numérico irredutível como um semigrupo numérico que não pode ser expresso como intersecção de dois semigrupos numéricos que o contenham propriamente. A finalizar este trabalho, estudamos os semigrupos numéricos irredutíveis e obtemos a decomposição de um semigrupo numérico em irredutíveis. ABSTRACT: A numerical semigroup is a submonoid of (N, +) such that its complement of N is finite. ln this work we study some invariants of a numerical semigroup S such as: multiplicity, embedding dimension, Frobenius number, gaps and Apéry set of S. We characterize a minimal presentation of a numerical semigroup S and describe an algorithmic procedure which allows us to compute a minimal presentation of S. We define an irreducible numerical semigroup as a numerical semigroup that cannot be expressed as the intersection of two numerical semigroups properly containing it. Concluding this work, we study and characterize irreducible numerical semigroups, and describe methods for computing decompositions of a numerical semigroup into irreducible numerical semigroups.

Stable isotope and multi-analytical investigation of Monte da Cegonha: A Late Antiquity population in southern Portugal

This study presents for the first time the diet of a Late Antiquity population in southern Portugal (Civitas of Pax Julia), from the Roman villa of Monte da Cegonha (predominantly 7th century CE). Stable isotope analysis (δ13C, δ15N, δ18O, 87Sr/86Sr) of human and faunal bone collagen and apatite was conducted in order to understand the influence of Roman subsistence strategies on the way of life of rural inhabitants of the area of Pax Julia and to explore their diet (types of ingested plants, amount of animal resources, terrestrial versus marine resources). X-ray diffraction (XRD) and Fourier transform infra-red spectroscopy (FTIR) analyses were used to determine the degree of bone diagenesis and assess the reliability of the bone stable isotopic composition for palaeodietary reconstruction. Anthropological analysis revealed a cariogenic diet, rich in starchy food and carbohydrates, in at least in two individuals based on the frequency of dental caries. Collagen and apatite carbon isotopic analysis suggested that C3 plants were the basis of the population's diet, complemented with some terrestrial meat and its by-products as reflected by the observed bone collagen nitrogen isotopic composition. Moreover, whilst the fairly low apatite-collagen spacing recorded in some skeletons (at around 4‰) may have been due to freshwater organisms intake, the relatively low nitrogen values observed indicate that this consumption did not occur very often, unless in the form of fresh fish of low trophic level or fish sauces. There were no significant differences in isotopic values depending on gender or burial type. Strontium and oxygen isotopic composition of bone apatite revealed a sedentary community, with the exception of a male individual who probably did not spend his childhood in Monte da Cegonha.