Classifying equivalence relations in the Ershov hierarchy.

Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility ⩽ c . This gives rise to a rich degree structure. In this paper, we lift the study of c-degrees to the Δ 2 0 case. In doing so, we rely on the Ershov hierarchy. For any notation a for a non-zero computable ordinal, we prove several algebraic properties of the degree structure induced by ⩽ c on the Σ a - 1 \ Π a - 1 equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of c-degrees.

Pomology observations, morphometric analysis, ultrastructural study and allelic profiles of "olivastra Seggianese" endocarps from ancient olive trees (Olea europaea L.).

Preliminary studies of historical sources and remote sensing were used to identify ancient olive trees near archaeological sites and heritage buildings in the Orcia Valley (Siena, Italy). Distinctive characters were assessed by traditional pomological observation. Trees with similar characters were selected on the basis of the features of endocarps, the only structure that survives aerobic deterioration and conserves useful botanical information for centuries. Non-invasive morphometric analysis of endocarp size and shape established morphological variations in individuals of different populations. Plastid organization in the endocarp and location of DNA in the endocarp tegument were detected by morphological and ultrastructural observations using light and electron microscopy. Cytoplasmic markers with high polymorphism were used to test similarity of endocarp and leaf DNA within individuals and to confirm low variability and minimal divergence between individuals. The ancient trees studied showed the same allelic profiles and therefore belonged to a distinct cultivar. The traditional pomological descriptions of the trees, leaves and fruits, morphometric analysis of size, and shape elliptic Fourier analysis of endocarp outline, ultrastructural observations and allelic profiles of endocarp tegument delineated the general species-specific qualities of the cultivar "olivastra Seggianese" of the Orcia Valley.