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A classical result of R.P. Dilworth states that every finite distributive lattice D can be represented as the congruence lattice of a finite lattice L. A sharper form was published in G. Grätzer and E.T. Schmidt in 1962, adding the requirement that all congruences in L be principal. Another variant, published in 1998 by the authors and E.T. Schmidt, constructs a planar semimodular lattice L. In this paper, we merge these two results: we construct L as a planar semimodular lattice in which all congruences are principal. This paper relies on the techniques developed by the authors and E.T. Schmidt in the 1998 paper.

Following Grätzer and Knapp, 2009, a planar semimodular lattice L is rectangular, if the left boundary chain has exactly one doubly-irreducible element, cl, and the right boundary chain has exactly one doubly-irreducible element, cr, and these elements are complementary.

A 1955 result of J. Jakubík states that for the prime intervals p and q of a finite lattice, con(p) ≥ con(q) iff p is congruence-projective to q (via intervals of arbitrary size). The problem is how to determine whether con(p) ≥ con(q) involving only prime intervals. Two recent papers approached this problem in different ways. G. Czédli’s used trajectories for slim rectangular lattices-a special subclass of slim, planar, semimodular lattices. I used the concept of prime-projectivity for arbitrary finite lattices. In this note I show how my approach can be used to reprove Czédli’s result and generalize it to arbitrary slim, planar, semimodular lattices.

Background and Aims: Both plants and animals display considerable variation in their phe- notypic traits as they grow. This variation helps organisms to adapt to specific challenges at different stages of development. Masting, the variable and synchronized seed production across years by a population of plants, is a common reproductive strategy in perennial plants that can enhance reproductive efficiency through increasing pollination efficiency and decreasing seed predation. Masting represents a population-level phenomenon generated from individual plant behaviors. While the developmental trajectory of individual plants influences their masting be- havior, the translation of such changes into benefits derived from masting remains unexplored., Methods and Key Results: We used 43 years of seed production monitoring in European beech (Fagus sylvatica) to address that gap. The largest improvements in reproductive efficiency from masting happen in the largest trees. Masting leads to a 48-fold reduction in seed predation in large, compared to 28-fold in small trees. Masting yields an 6-fold increase in pollination efficiency in large, compared to 2.5-fold in small trees. Paradoxically, although the largest trees show the biggest reproductive efficiency benefits from masting, large trees mast less strongly than small trees., Conclusions: That apparently suboptimal allocation of effort across years by large plants may be a consequence of anatomical constraints or bet-hedging. Ontogenetic shifts in individual mast- ing behavior and associated variable benefits have implications for the reproductive potential of plant populations as their age distribution changes, with applications in plant conservation and management., (© The Author(s) 2024. Published by Oxford University Press on behalf of the Annals of Botany Company. All rights reserved. For commercial re-use, please contact reprints@oup.com for reprints and translation rights for reprints. All other permissions can be obtained through our RightsLink service via the Permissions link on the article page on our site—for further information please contact journals.permissions@oup.com.)

Many perennial plants show mast seeding, characterized by synchronous and highly variable reproduction across years. We propose a general model of masting, integrating proximate factors (environmental variation, weather cues, and resource budgets) with ultimate drivers (predator satiation and pollination efficiency). This general model shows how the relationships between masting and weather shape the diverse responses of species to climate warming, ranging from no change to lower interannual variation or reproductive failure. The role of environmental prediction as a masting driver is being reassessed; future studies need to estimate prediction accuracy and the benefits acquired. Since reproduction is central to plant adaptation to climate change, understanding how masting adapts to shifting environmental conditions is now a central question., Competing Interests: Declaration of interests No competing interests to declare., (Copyright © 2024 The Authors. Published by Elsevier Ltd.. All rights reserved.)

The fundamental trade-off between current and future reproduction has long been considered to result in a tendency for species that can grow large to begin reproduction at a larger size. Due to the prolonged time required to reach maturity, estimates of tree maturation size remain very rare and we lack a global view on the generality and the shape of this trade-off. Using seed production from five continents, we estimate tree maturation sizes for 486 tree species spanning tropical to boreal climates. Results show that a species' maturation size increases with maximum size, but in a non-proportional way: the largest species begin reproduction at smaller sizes than would be expected if maturation were simply proportional to maximum size. Furthermore, the decrease in relative maturation size is steepest in cold climates. These findings on maturation size drivers are key to accurately represent forests' responses to disturbance and climate change., (© 2024 John Wiley & Sons Ltd.)

Climate change effects on tree reproduction are poorly understood, even though the resilience of populations relies on sufficient regeneration to balance increasing rates of mortality. Forest-forming tree species often mast, i.e. reproduce through synchronised year-to-year variation in seed production, which improves pollination and reduces seed predation. Recent observations in European beech show, however, that current climate change can dampen interannual variation and synchrony of seed production and that this masting breakdown drastically reduces the viability of seed crops. Importantly, it is unclear under which conditions masting breakdown occurs and how widespread breakdown is in this pan-European species. Here, we analysed 50 long-term datasets of population-level seed production, sampled across the distribution of European beech, and identified increasing summer temperatures as the general driver of masting breakdown. Specifically, increases in site-specific mean maximum temperatures during June and July were observed across most of the species range, while the interannual variability of population-level seed production (CVp) decreased. The declines in CVp were greatest, where temperatures increased most rapidly. Additionally, the occurrence of crop failures and low seed years has decreased during the last four decades, signalling altered starvation effects of masting on seed predators. Notably, CVp did not vary among sites according to site mean summer temperature. Instead, masting breakdown occurs in response to warming local temperatures (i.e. increasing relative temperatures), such that the risk is not restricted to populations growing in warm average conditions. As lowered CVp can reduce viable seed production despite the overall increase in seed count, our results warn that a covert mechanism is underway that may hinder the regeneration potential of European beech under climate change, with great potential to alter forest functioning and community dynamics., (© 2024 The Authors. Global Change Biology published by John Wiley & Sons Ltd.)

The benefits of masting (volatile, quasi-synchronous seed production at lagged intervals) include satiation of seed predators, but these benefits come with a cost to mutualist pollen and seed dispersers. If the evolution of masting represents a balance between these benefits and costs, we expect mast avoidance in species that are heavily reliant on mutualist dispersers. These effects play out in the context of variable climate and site fertility among species that vary widely in nutrient demand. Meta-analyses of published data have focused on variation at the population scale, thus omitting periodicity within trees and synchronicity between trees. From raw data on 12 million tree-years worldwide, we quantified three components of masting that have not previously been analysed together: (i) volatility, defined as the frequency-weighted year-to-year variation; (ii) periodicity, representing the lag between high-seed years; and (iii) synchronicity, indicating the tree-to-tree correlation. Results show that mast avoidance (low volatility and low synchronicity) by species dependent on mutualist dispersers explains more variation than any other effect. Nutrient-demanding species have low volatility, and species that are most common on nutrient-rich and warm/wet sites exhibit short periods. The prevalence of masting in cold/dry sites coincides with climatic conditions where dependence on vertebrate dispersers is less common than in the wet tropics. Mutualist dispersers neutralize the benefits of masting for predator satiation, further balancing the effects of climate, site fertility and nutrient demands., (© 2023. The Author(s), under exclusive licence to Springer Nature Limited.)

In Europe, forest management has controlled forest dynamics to sustain commodity production over multiple centuries. Yet over-regulation for growth and yield diminishes resilience to environmental stress as well as threatens biodiversity, leading to increasing forest susceptibility to an array of disturbances. These trends have stimulated interest in alternative management systems, including natural dynamics silviculture (NDS). NDS aims to emulate natural disturbance dynamics at stand and landscape scales through silvicultural manipulations of forest structure and landscape patterns. We adapted a "Comparability Index" (CI) to assess convergence/divergence between natural disturbances and forest management effects. We extended the original CI concept based on disturbance size and frequency by adding the residual structure of canopy trees after a disturbance as a third dimension. We populated the model by compiling data on natural disturbance dynamics and management from 13 countries in Europe, covering four major forest types (i.e., spruce, beech, oak, and pine-dominated forests). We found that natural disturbances are highly variable in size, frequency, and residual structure, but European forest management fails to encompass this complexity. Silviculture in Europe is skewed toward even-aged systems, used predominately (72.9% of management) across the countries assessed. The residual structure proved crucial in the comparison of natural disturbances and silvicultural systems. CI indicated the highest congruence between uneven-aged silvicultural systems and key natural disturbance attributes. Even so, uneven-aged practices emulated only a portion of the complexity associated with natural disturbance effects. The remaining silvicultural systems perform poorly in terms of retention compared to tree survivorship after natural disturbances. We suggest that NDS can enrich Europe's portfolio of management systems, for example where wood production is not the primary objective. NDS is especially relevant to forests managed for habitat quality, risk reduction, and a variety of ecosystem services. We suggest a holistic approach integrating NDS with more conventional practices., (© 2022 The Ecological Society of America.)

The relationships that control seed production in trees are fundamental to understanding the evolution of forest species and their capacity to recover from increasing losses to drought, fire, and harvest. A synthesis of fecundity data from 714 species worldwide allowed us to examine hypotheses that are central to quantifying reproduction, a foundation for assessing fitness in forest trees. Four major findings emerged. First, seed production is not constrained by a strict trade-off between seed size and numbers. Instead, seed numbers vary over ten orders of magnitude, with species that invest in large seeds producing more seeds than expected from the 1:1 trade-off. Second, gymnosperms have lower seed production than angiosperms, potentially due to their extra investments in protective woody cones. Third, nutrient-demanding species, indicated by high foliar phosphorus concentrations, have low seed production. Finally, sensitivity of individual species to soil fertility varies widely, limiting the response of community seed production to fertility gradients. In combination, these findings can inform models of forest response that need to incorporate reproductive potential., (© 2022. The Author(s).)

Significant gaps remain in understanding the response of plant reproduction to environmental change. This is partly because measuring reproduction in long-lived plants requires direct observation over many years and such datasets have rarely been made publicly available. Here we introduce MASTREE+, a data set that collates reproductive time-series data from across the globe and makes these data freely available to the community. MASTREE+ includes 73,828 georeferenced observations of annual reproduction (e.g. seed and fruit counts) in perennial plant populations worldwide. These observations consist of 5971 population-level time-series from 974 species in 66 countries. The mean and median time-series length is 12.4 and 10 years respectively, and the data set includes 1122 series that extend over at least two decades (≥20 years of observations). For a subset of well-studied species, MASTREE+ includes extensive replication of time-series across geographical and climatic gradients. Here we describe the open-access data set, available as a.csv file, and we introduce an associated web-based app for data exploration. MASTREE+ will provide the basis for improved understanding of the response of long-lived plant reproduction to environmental change. Additionally, MASTREE+ will enable investigation of the ecology and evolution of reproductive strategies in perennial plants, and the role of plant reproduction as a driver of ecosystem dynamics., (© 2022 The Authors. Global Change Biology published by John Wiley & Sons Ltd.)

The timing of seed production and release is highly relevant for successful plant reproduction. Ecological disturbances, if synchronized with reproductive effort, can increase the chances of seeds and seedlings to germinate and establish. This can be especially true under variable and synchronous seed production (masting). Several observational studies have reported worldwide evidence for co-occurrence of disturbances and seed bumper crops in forests. Here, we review the evidence for interaction between disturbances and masting in global plant communities; we highlight feedbacks between these two ecological processes and posit an evolutionary pathway leading to the selection of traits that allow trees to synchronize seed crops with disturbances. Finally, we highlight relevant questions to be tested on the functional and evolutionary relationship between disturbances and masting. This article is part of the theme issue 'The ecology and evolution of synchronized seed production in plants'.