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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.

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 Elsevier Ltd. All rights reserved.)

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.)

In disturbance-prone ecosystems, fitness consequences of plant reproductive strategies are often determined by the relative timing of seed production and disturbance events, but the role of disturbances as proximate drivers of seed production has been overlooked. We use long-term data on seed production in Quercus chapmanii , Q. geminata and Q. inopina , rhizomatous oaks found in south central Florida's oak scrub, to investigate the role of fire history and its interaction with weather in shaping acorn production and its synchrony . Acorn production increased with the time since last fire, combined with additive or interactive effects of spring precipitation (+) or drought (-). Furthermore, multiple matrix regression models revealed that ramet pairs with shared fire history were more synchronous in seed production than ones that burned in different years. Long-term trends suggest that increasingly drier spring weather, in interaction with fire frequency, may drive a decline of seed production. Such declines could affect the community of acorn-reliant vertebrates in the Florida scrub, including endangered Florida scrub-jays ( Aphelocoma coerulescens ). These results illustrate that fire can function as a proximate driver of seed production in mast-seeding species, highlighting the increasingly recognized importance of interactions among reproductive strategies and disturbance regimes in structuring plant populations and communities. This article is part of the theme issue 'The ecology and evolution of synchronized seed production in plants'.

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'.

Mast seeding, the synchronized interannual variation in seed production of trees, is a well-known bottom-up driver for population densities of granivorous forest rodents. Such demographic effects also affect habitat preferences of the animals: After large seed production events, reduced habitat selectivity can lead to spillover from forest patches into adjacent alpine meadows or clear-cuts, as has been reported for human-impacted forests. In unmanaged, primeval forests, however, gaps created by natural disturbances are typical elements, yet it is unclear whether the same spillover dynamics occur under natural conditions. To determine whether annual variation in seed production drives spillover effects in naturally formed gaps, we used 14 years of small mammal trapping data combined with seed trap data to estimate population densities of Apodemus spp. mice and bank voles ( Myodes glareolus ) on 5 forest sites with differing disturbance history. The study sites, located in a forest dominated by European beech ( Fagus sylvatica ), Norway spruce ( Picea abies ), and silver fir ( Abies alba ), consisted of two primeval forest sites with small canopy gaps, two sites with larger gaps (after an avalanche event and a windthrow event), and a managed forest stand with closed canopy as a control. Hierarchical Bayesian N-mixture models revealed a strong influence of seed rain on small rodent abundance, which were site-specific for M. glareolus but not for Apodemus spp. Following years of moderate or low seed crop, M. glareolus avoided open habitat patches but colonized those habitats in large numbers after full mast events, suggesting that spillover events also occur in unmanaged forests, but not in all small rodents. The species- and site-specific characteristics of local density responding to food availability have potentially long-lasting effects on forest gap regeneration dynamics and should be addressed in future studies., Competing Interests: None declared., (© 2021 The Authors. Ecology and Evolution published by John Wiley & Sons Ltd.)

Herbivores are constitutive elements of most terrestrial ecosystems. Understanding effects of herbivory on ecosystem dynamics is thus a major, albeit challenging task in community ecology. Effects of mammals on plant communities are typically explored by comparing plant densities or diversity in exclosure experiments. This might over-estimate long-term herbivore effects at community levels as early life stage mortality is driven by a multitude of factors. Addressing these challenges, we established a set of 100 pairs of ungulate exclosures and unfenced control plots (25 m 2 ) in mixed montane forests in the Alps in 1989 covering a forest area of 90 km 2 . Investigations ran until 2013. Analogous to the gap-maker-gap-filler approach, dynamically recording the height of the largest trees per tree species in paired plots with and without exclosures might allow for assessing herbivore impacts on those individuals with a high probability of attaining reproductive stages. We thus tested if recording maximum heights of regenerating trees would better reflect effects of ungulate herbivory on long-term dynamics of tree regeneration than recording of stem density, and if species dominance patterns would shift over time. For quantifying the effects of ungulate herbivory simultaneously at community and species level we used principle response curves (PRC). PRCs yielded traceable results both at community and species level. Trajectories of maximum heights yielded significant results contrary to trajectories of total stem density. Response patterns of tree species were not uniform over time: e.g., both Norway spruce and European larch switched in their response to fencing. Fencing explained about 3% of the variance of maximum tree heights after nine years but increased to about 10% after 24 years thus confirming the importance of long-term surveys. Maximum height dynamics of tree species, addressed in our study, can thus reflect local dominance of tree species via asymmetric plant competition. Such effects, both within and among forest patches, can accrue over time shaping forest structure and composition.

In the central and northern highlands of Ethiopia, native forest and forest biodiversity is almost confined to sacred groves associated with churches. Local communities rely on these 'church forests' for essential ecosystem services including shade and fresh water but little is known about their region-wide distribution and conservation value. We (1) performed the first large-scale spatially-explicit assessment of church forests, combining remote-sensing and field data, to assess the number of forests, their size, shape, isolation and woody plant species composition, (2) determined their plant communities and related these to environmental variables and potential natural vegetation, (3) identified the main challenges to biodiversity conservation in view of plant population dynamics and anthropogenic disturbances, and (4) present guidelines for management and policy. The 394 forests identified in satellite images were on average ~2ha in size and generally separated by ~2km from the nearest neighboring forest. Shape complexity, not size, decreased from the northern to the central highlands. Overall, 148 indigenous tree, shrub and liana species were recorded across the 78 surveyed forests. Patch α-diversity increased with mean annual precipitation, but typically only 25 woody species occurred per patch. The combined results showed that >50% of tree species present in tropical northeast Africa were still present in the 78 studied church forests, even though individual forests were small and relatively species-poor. Tree species composition of church forests varied with elevation and precipitation, and resembled the potential natural vegetation. With a wide distribution over the landscape, these church forests have high conservation value. However, long-term conservation of biodiversity of individual patches and evolutionary potential of species may be threatened by isolation, small sizes of tree species populations and disturbance, especially when considering climate change. Forest management interventions are essential and should be supported by environmental education and other forms of public engagement., (Copyright © 2016 Elsevier B.V. All rights reserved.)