Partial arithmetical data types of rational numbers and their equational specification

Upon adding division to the operations of a field we obtain a meadow. It is conventional toview division in a field as a partial function, which complicates considerably its algebra andlogic. But partiality is one out of a plurality of possible design decisions regarding division.Upon adding a partial division function ÷ to a field Q of rational numbers we obtain apartial meadow Q (÷) of rational numbers that qualifies as a data type. Partial data typesbring problems for specifying and programming that have led to complicated algebraicand logical theories – unlike total data types. We discuss four different ways of providingan algebraic specification of this important arithmetical partial data type Q (÷) via thealgebraic specification of a closely related total data type. We argue that the specificationmethod that uses a common meadow of rational numbers as the total algebra is themost attractive and useful among these four options. We then analyse the problem ofequality between expressions in partial data types by examining seven notions of equalitythat arise from our methods alone. Finally, based on the laws of common meadows, wepresent an equational calculus for working with fracterms that is of general interest outsideprogramming theory

A Complete Finite Equational Axiomatisation of the Fracterm Calculus for Common Meadows

We analyse abstract data types that model numerical structures with a concept of error. Specifically, we focus on arithmetic data types that contain an error flag $\bot$ whose main purpose is to always return a value for division. To rings and fields we add a division operator $x/y$ and study a class of algebras called \textit{common meadows} wherein $x/0 = \bot$. The set of equations true in all common meadows is named the \textit{fracterm calculus of common meadows}. We give a finite equational axiomatisation of the fracterm calculus of common meadows and prove that it is complete and that the fracterm calculus is decidable

Mediterranean grassland succession as an indicator of changes in ecosystem biodiversity and functionality

The abandonment of agricultural lands triggers a secondary succession of plant species which implies important changes in soil quality. Annual Mediterranean grasslands are known to be persistent on abandoned agriculture lands in the western Mediterranean. We used plant taxonomic and functional approaches to determine the role of Mediterranean grasslands as an indicator of changes in ecosystem biodiversity and functionality. We tested the hypothesis that Mediterranean grasslands are a suitable model for monitoring biodiversity and soil fertility in a secondary succession. Soil and vegetation features on 21 permanent plots were monitored in 2016 and 2020. Numerical classifications based on floristic composition showed two different plant communities independently of the sampling year: early-stage grasslands in the first post-abandonment decade and late-stage grasslands after the first post-abandonment decade. Generalized linear model and redundancy analysis also revealed differences in growth forms, functional traits and soil functionality between communities. Late-stage grasslands was characterized by enriched bryophyte coverage and an impoverishment in hemicryptophytes and plant latex segregators growing on soils with a higher hydrolase enzyme activity and TOC content compared to early-stage grassland. Our results suggest that annual Mediterranean grasslands growing on siliceous soils denoting a mature-stage succession, and floristically characterized by the symbiont plant with Ascomycota, Tuberaria guttata, and a high bryophyte cover, are worthy of recognition for conservation