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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.

Cylindrocarpon-like fungi are globally distributed plant pathogens and have a wide range of host species. However, very little is known about the species that live in the topsoil of forests and their potential role in reducing the natural regeneration of tree species, particularly of forest trees that produce abundant fruit only once every few years. To enhance our understanding of the species that inhabit the topsoil, we studied the diversity and pathogenicity of Cylindrocarpon-like fungi inhabiting the litter in old-growth mixed-beech forests in the Carpathians (Poland) and in the Alps (Austria), and in a managed beech stand in the Krakowsko-Częstochowska Highland (Poland). The fungi inhabiting the beech litter were investigated using beechnuts and pine seedlings as bait. Isolates were identified based on morphology and DNA sequencing. The pathogenicity of the most common species was investigated by inoculating beech germinants. A wide range of Cylindrocarpon-like fungi were associated with the beech litter: 718 cultures representing 12 species were isolated. Five taxa were identified down to species level: namely Ilyonectria crassa, I. pseudodestructans, I. rufa, Neonectria candida and N. obtusispora, and seven species were identified to genus level ( Neonectria or Ilyonectria species). Ilyonectria destructans, which is considered to be the sexual morph of ' Cylindrocarpon destructans', was not found. There were qualitative and quantitative differences between the different forest sites in terms of Neonectria and Ilyonectria species composition in beech litter. The isolation frequency and species richness of Cylindrocarpon-like fungi were greatest in beech litter taken from old-growth mixed-beech forests. Neonectria and Ilyonectria species were capable of killing beech germinants, suggesting that they may play a negative role in natural beech regeneration. [ABSTRACT FROM AUTHOR]

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.

Temperate old-growth forests are known to have ecological characteristics distinct from younger forests, but these have been poorly described for the remaining old-growth Picea abies-Abies alba forests in the eastern Carpathian mountains. In addition, recent studies suggest that old-growth forests may be more significant carbon sinks than previously recognized. This has stimulated interest in quantifying aboveground carbon stocks in primary forest systems. We investigated the structural attributes and aboveground biomass in two remnant old-growth spruce-fir stands and compared these against a primary (never logged) mature reference stand. Our sites were located in the Gorgany Nature Reserve in western Ukraine. Overstory data were collected using variable radius plots; coarse woody debris was sampled along line intercept transects. Differences among sites were assessed using non-parametric statistical analyses. Goodness-of-fit tests were used to evaluate the form of diameter distributions. The results strongly supported the hypothesis that old-growth temperate spruce-fir forests have greater structural complexity compared to mature forests, including higher densities of large trees, more complex horizontal structure, and elevated aboveground biomass. The late-successional sites we sampled exhibited rotated sigmoid diameter distributions; these may reflect natural disturbance dynamics. Old-growth Carpathian spruce-fir forests store on average approximately 155-165 Mg ha-1 of carbon in aboveground tree parts alone. This is approximately 50% higher than mature stands. Given the scarcity of primary spruce-fir forests in the Carpathian region, remaining stands have high conservation value, both as habitat for late-successional species and as carbon storage reservoirs. [ABSTRACT FROM AUTHOR]

Abstract: In a temperate mixed conifer forest in the Bhutan Himalayas, we investigated the effects of cattle grazing on conifer (Tsuga dumosa, Pinus wallichiana, Picea spinulosa, Abies densa) seedling density, growth and microsite of recruitment under dense cover of the bamboo Yushania microphylla, using exclosures over a period of 9 years after group selection harvest. Increasing bamboo competition over time following canopy opening successfully prevented seedling recruitment in ungrazed plots, while recruitment in grazed plots was continuous. Reduction of bamboo height through grazing facilitated recruitment of all tree species, particularly T. dumosa, mainly through increased light interception on the forest floor. Tree species composition of seedlings and the overstory did not differ in ungrazed plots, while in grazed plots we observed a shift towards dominance of T. dumosa. Growth rates of T. dumosa and P. spinulosa were higher in grazed plots as compared to ungrazed plots. In grazed plots, recruitment of T. dumosa was concentrated on moss, which might have prevented desiccation of the small-seeded species after germination. We propose that controlled grazing might facilitate natural regeneration after logging in mixed conifer forests of central Bhutan with dense Y. microphylla bamboo understory. [Copyright &y& Elsevier]

To assess the impact of a dense understory of the bamboo Yushania microphylla Munro on tree regeneration in monospecific Abies densa Griff. stands of the central Bhutan Himalayas, the age-class distributionof fir regeneration, the microsite preferences, and height growth aswell as the relationship between height of the bamboo, gap fraction,and tree seedling density were studied. Seedling densities were muchlower on sites with bamboo. Recruitment was more or less continuous,and there was no indication of overall synchronized single-cohort regeneration in bamboo plots. On sites with bamboo understory, the light regime at the forest floor is strongly influenced by the height of the bamboo. A large portion of the variance of tree seedling density could be explained by the height of the bamboo. The mortality of fir seedlings is considerably reduced at elevated microsites. Fir establishment on the forest floor is restricted to sites where bamboo density is low and light levels are higher. On sites with dense, unbrowsed bamboo, light levels are too low for long-term survival of fir seedlings, resulting in a lack of suppressed regeneration with minimal height growth. Because of the absence of this fraction, the average height increment of fir seedlings is higher on sites with bamboo. [ABSTRACT FROM AUTHOR]