The Cantor-Bernstein-Schröder theorem of the set theory was generalized by Sikorski and Tarski to $\sigma $-complete boolean algebras, and recently by several authors to other algebraic structures. In this paper we expose an abstract version which is applicable to algebras with an underlying lattice structure and such that the central elements of this lattice determine a direct decomposition of the algebra. Necessary and sufficient conditions for the validity of the Cantor-Bernstein-Schröder theorem for these algebras are given. These results are applied to obtain versions of the Cantor-Bernstein-Schröder theorem for $\sigma $-complete orthomodular lattices, Stone algebras, $BL$-algebras, $MV$-algebras, pseudo $MV$-algebras, Łukasiewicz and Post algebras of order $n$.
A field experiment involving two planting densities (83,333 and 166,666 plants per ha), two cropping systems (monoculture and mixed culture) and five cowpea [Vigna unguiculata L. (Walp.)] genotypes was conducted at Nietvoorbij (33°54S, 18°14E), Stellenbosch, South Africa, to select cowpea material with superior growth and water-use efficiency (WUE). The results showed significantly higher photosynthetic rates, stomatal conductance and transpiration in leaves of plants at low density and in monoculture due to greater chlorophyll (Chl) levels relative to those at high density and in mixed culture. As a result, C concentration in leaves and the amount of C, P, K, Ca, Mg, Fe, Cu, Zn, Mn, and B accumulated in shoots at low density and under monoculture were also much higher. Even though no marked differences in photosynthetic rates were found between and among the five cowpea genotypes, leaf C concentration and shoot C, P, K, Ca, Mg, Fe, Cu, Zn, Mn, and B contents differed considerably, with Sanzie exhibiting the highest C concentration and C, P, K, Ca, Mg, Fe, Cu, Zn, Mn, and B contents in shoots, followed by Bensogla and Omondaw, while ITH98-46 and TVu1509 had the lowest shoot concentration and contents of C, P, K, Ca, Mg, Fe, Cu, Zn, Mn, and B. WUE (calculated as photosynthate produced per unit water molecule transpired) was significantly greater in plants at low density and monoculture relative to those at high density and in mixed culture. Isotopic analysis revealed significant differences in δ13C values of sorghum [Sorghum bicolor L. (Moench.)] and cowpea, with higher δ13C values being obtained for plants at low density and in monoculture relative to those at high density or in mixed culture. The five cowpea genotypes also showed significant differences in δ13C values, with Sanzie exhibiting the most negative value (i.e. low WUE) and ITH98-46, the least negative δ13C value (i.e. high WUE). Whether measured isotopically or from gas-exchange studies, sorghum (a C4 species) exhibited much higher WUE relative to cowpea (a C3 species). Both correlation and regression analyses revealed a positive relationship between WUE from gas-exchange studies and δ13C values from isotopic analysis of cowpea and sorghum shoots. and J. H. J. R. Makoi, S. B. M. Chimphango, F. D. Dakora
We study the weak hereditary class $S_{w}(\mathcal K)$ of all weak subalgebras of algebras in a total variety $\mathcal K$. We establish an algebraic characterization, in the sense of Birkhoff’s HSP theorem, and a syntactical characterization of these classes. We also consider the problem of when such a weak hereditary class is weak equational.