In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent:

• Corollary.LetKbe a tame AEC with a monster model. Assume thatKis stable in a proper class of cardinals. The following are equivalent:

  1. (1) For all high-enough λ,Khas no long splitting chains.

  2. (2) For all high-enough λ, there exists a good λ-frame on a skeleton ofKλ.

  3. (3) For all high-enough λ,Khas a unique limit model of cardinality λ.

  4. (4) For all high-enough λ,Khas a superlimit model of cardinality λ.

  5. (5) For all high-enough λ, the union of any increasing chain of λ-saturated models is λ-saturated.

  6. (6) There exists μ such that for all high-enough λ,Kis (λ,μ) -solvable.

This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.

Studies were conducted to determine the growth responses of rice to PRE application of oxadiazon and POST application of bispyribac-sodium. Oxadiazon at 1.0 and 1.5 kg ha−1 and bispyribac-sodium at 0.030 and 0.045 kg ha−1 were applied to four rice varieties (‘IR64’, ‘IR72’, ‘RC09’, and ‘RC18’), which were grown in saturated and aerobic (30% of saturation) soils. Control treatments, where no herbicides were applied, were also included in the study. Shoot and root biomass, and height of rice plants were measured 14 d after application. Phytotoxic effects for both herbicides, including reduced shoot and root biomass, were consistent in all varieties. Rice phytotoxicity symptoms were greater when herbicides were applied to saturated than to aerobic soils. Oxadiazon at 1.0 kg ha−1 reduced rice shoot biomass by 22 to 36% in aerobic condition, and 43 to 56% in saturated condition when compared with the control. Bispyribac-sodium reduced rice shoot biomass by 9 to 17% at 0.030 kg ha−1 in aerobic soil and 23 to 37% in saturated soil. The results of this study suggest that soil water content is an important factor influencing herbicide phytotoxicity in rice, and its influence warrants further research to improve understanding of physiology of phytotoxicity to minimize the effects of these herbicides on crop production.

We determine which permutative varieties are saturated and classify all nontrivial permutation identities for the class of all globally idempotent semigroups.

We establish a necessary condition (E) for a semigroup variety to be closed under the taking of epimorphisms and a necessary condition (S) for a variety to consist entirely of saturated semigroups. Condition (S) is shown to be sufficient for heterotypical varieties and a stronger condition (S′) is shown to be sufficient for homotypical varieties.