Singly generated quasivarieties and residuated structures

A quasivariety K of algebras has the joint embedding property (JEP) iff it is generated by a single algebra A. It is structurally complete iff the free countably generated algebra in K can serve as A. A consequence of this demand, called "passive structural completeness" (PSC), is that the nontrivial members of K all satisfy the same existential positive sentences. We prove that if K is PSC then it still has the JEP, and if it has the JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if K is relatively semisimple then it is generated by one K-simple algebra. It is a minimal quasivariety if, moreover, it is PSC but fails to unify some finite set of equations. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC iff its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics

Four Variables Suffice

What I wish to propose in the present paper is a new form of “career induction” for ambitious young logicians. The basic problem is this: if we look at the n-variable fragments of relevant propositional logics, at what point does undecidability begin? Focus, to be definite, on the logic R. John Slaney showed that the 0-variable fragment of R (where we allow the sentential con- stants t and f) contains exactly 3088 non-equivalent propositions, and so is clearly decidable. In the opposite direction, I claimed in my paper of 1984 that the five variable fragment of R is undecidable. The proof given there was sketchy (to put the matter charitably), and a close examination reveals that although the result claimed is true, the proof given is incorrect. In the present paper, I give a detailed and (I hope) correct proof that the four variable fragments of the principal relevant logics are undecidable. This leaves open the question of the decidability of the n-variable fragments for n = 1, 2, 3. At what point does undecidability set in

Four Variables Suffice

What I wish to propose in the present paper is a new form of “career induction” for ambitious young logicians. The basic problem is this: if we look at the n-variable fragments of relevant propositional logics, at what point does undecidability begin? Focus, to be definite, on the logic R. John Slaney showed that the 0-variable fragment of R (where we allow the sentential con- stants t and f) contains exactly 3088 non-equivalent propositions, and so is clearly decidable. In the opposite direction, I claimed in my paper of 1984 that the five variable fragment of R is undecidable. The proof given there was sketchy (to put the matter charitably), and a close examination reveals that although the result claimed is true, the proof given is incorrect. In the present paper, I give a detailed and (I hope) correct proof that the four variable fragments of the principal relevant logics are undecidable. This leaves open the question of the decidability of the n-variable fragments for n = 1, 2, 3. At what point does undecidability set in

Singly generated quasivarieties and residuated structures

Please read abstract in the article.H2020 Marie Skłodowska-Curie Actions; DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa and National Research Foundation of South Africa.https://onlinelibrary.wiley.com/journal/15213870hj2021Mathematics and Applied Mathematic

Varieties of De Morgan monoids : covers of atoms

The variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4{element De Morgan monoids C4 and D4, where C4 is the only nontrivial 0{generated algebra onto which nitely subdirectly irreducible De Morgan monoids may be mapped by non-injective homomorphisms. The homomorphic pre-images of C4 within DMM (together with the trivial De Morgan monoids) constitute a proper quasivariety, which is shown to have a largest subvariety U. The covers of the variety V(C4) within U are revealed here. There are just ten of them (all nitely generated). In exactly six of these ten varieties, all nontrivial members have C4 as a retract. In the varietal join of those six classes, every subquasivariety is a variety|in fact, every nite subdirectly irreducible algebra is projective. Beyond U, all covers of V(C4) [or of V(D4)] within DMM are discriminator varieties. Of these, we identify in nitely many that are nitely generated, and some that are not. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids.The European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant, RVO 67985807 and by the CAS-ICS postdoctoral fellowship, the National Research Foundation of South Africa and DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa.https://www.cambridge.org/core/journals/review-of-symbolic-logic2021-06-01am2021Mathematics and Applied Mathematic

Semilinear De Morgan monoids and epimorphisms

DATA AVAILABILITY : Data sharing not applicable to this article as datasets were neither generated nor analysed.A representation theorem is proved for De Morgan monoids that are (i) semilinear, i.e., subdirect products of totally ordered algebras, and (ii) negatively generated, i.e., generated by lower bounds of the neutral element. Using this theorem, we prove that the De Morgan monoids satisfying (i) and (ii) form a variety—in fact, a locally finite variety. We then prove that epimorphisms are surjective in every variety of negatively generated semilinear De Morgan monoids. In the process, epimorphism-surjectivity is established for several other classes as well, including the variety of all semilinear idempotent commutative residuated lattices and all varieties of negatively generated semilinear Dunn monoids. The results settle natural questions about Beth-style definability for a range of substructural logics.The Operational Programme Research, Development and Education of the Ministry of Education, Youth and Sports of the Czech Republic, the EU and in part by the National Research Foundation of South Africa. Open access funding provided by University of Pretoria.https://link.springer.com/journal/12hj2024Mathematics and Applied MathematicsNon

Elucidating the role of Glycoside Hydrolase 25 in microbial antagonsim

Plant organs are colonized by microorganisms, ranging from beneficial, commensals and pathogenic, which interact among themselves to establish a community. Rhizosphere microbiota is extensively characterized regarding their roles in protection of host plant against pathogenic microbes and in overall plant growth and development. However, the phyllosphere is also enriched with microbial interactions, aiding in plant immunity and protection. In this study, it was shown how a basidiomycete yeast Moesziomyces bullatus ex Albugo on Arabidopsis (MbA) antagonizes the white rust pathogen Albugo laibachii by expressing Glycoside Hydrolase 25 (GH25) protein on Arabidopsis thaliana. Recombinantly produced MbA_GH25 also reduced A. laibachii growth in-planta, identifying the hydrolase gene to be a major effector of MbA mediated antagonism. GH25 proteins are present in various plant�associated fungi, however, their mechanism of action is unknown. To elucidate the mechanism behind MbA_GH25 activity, two strategies were employed: firstly, testing purified MbA_GH25 as an elicitor of plant immunity and secondly, analyzing the impact of MbA_GH25 on bacterial associations of A. laibachii. The potential antagonistic activity of MbA_GH25 towards the oomycetes Hyaloperenospora arabidopsidis and Phytophthora infestans have been investigated as well. Alongside the role of GH25 in related smut pathogen, U. maydis has been explored in this study to understand the impact of the hydrolase gene expression on fungal pathogenicity. Taken together, we functionally characterize the role of fungal GH25 proteins to deepen our understanding of microbial interactions in the phyllosphere and carve a path for the development of newer antimicrobials for plant protection

Trace Hydrogen in Minerals

Trace hydrogen in minerals most frequently occurs bonded to oxygen. The resulting water and hydroxyl (OH-) affect and play a role in a variety of mineral properties and reactions. This thesis examines the occurrence of trace hydrogen in nominally anhydrous minerals, the mechanisms by which trace hydrogen participates in reactions and controls properties, and the changes that occur in hydrogen speciation and siting as a function of temperature. The principal tool used in this study is infrared (IR) spectroscopy because of its sensitivity to the highly polar O-H bond, yielding quantitative information on concentration, and symmetry, speciation, and siting information. The speciation of trace hydrogen in garnet and low temperature natural and synthetic quartz is examined in detail. In garnet hydrogen occurs as the hydrogarnet substitution, four hydroxyl groups replacing a silicate tetrahedron. This substitution is extremely common among natural garnets. Concentrations range from 0.05 to 0.20 wt. % (as H2O) in garnets from most occurrences, including garnets from the mantle. This trace hydrogen is truly dissolved. The hydrogen found in natural and synthetic quartz formed at low temperature can occur as either hydroxyl or molecular water. The molecular water is the active participant in hydrolytic weakening of quartz, but it is not truly dissolved. It occurs as small groups of molecules (approximately 5 to 200) which were trapped during rapid growth. Two properties of minerals affected by trace hydrogen are strength and radiation response. Molecular water may be responsible for weakening of other minerals as well as quartz. Both water and hydroxyl participate in radiation response of minerals. In metamict zircon, water stabilizes local charge imbalance formed when bonds are broken. Water enters the crystal after a threshold of damage occurs, and reacts with broken bonds to form hydroxyl groups. These must reform molecular water and be expelled before recrystallization occurs during heating. In quartz, molecular water is strongly correlated with the formation of citrine color during irradiation, but inhibits the formation of the amethyst color center Fe4+. Apparently molecular hydrogen forms during radiolysis of the water, and reduces the Fe4+. Several hydroxyl sites in topaz are strongly correlated with the formation of brown color upon irradiation. The unifying theme in all these reactions is the extreme mobility of hydrogen and the ease with which different oxygen-hydrogen species may be formed in silicates. The behavior of trace hydrogen at temperatures of geologic interest has been examined using high temperature infrared spectroscopy. Direct observations of speciation, concentration, and properties have been made up to 1200°C. In muscovite there is no change in hydrogen speciation or site up to the dehydration point, as expected. However, in cordierite and beryl water reversibly partitions into a gas-like state above 400°C, and the formation of this new state controls the dehydration behavior. In topaz, hydroxyl groups have been observed converting to new sites at temperatures above 500°C. In orthoclase feldspar, one type of molecular water dehydrates at 200°C, while a second type converts irreversibly to a new hydrous species above 600°C. There is no evidence for the existence of hydrogen species other than hydroxyl and water in silicate minerals. The hydrogarnet substitution (four hydroxyl groups in a tetrahedral configuration) is common in garnets and may be important in other orthosilicates. The most common hydrous species in nominally anhydrous silicates (aside from fluid inclusions and alteration) are: small groups of trapped water molecules; individual water molecules occupying voids in the structure of minerals; hydroxyl occurring in a charge balancing role such as AlO3OH substituting for SiO4; hydroxyl neutralizing substitutional atoms, e.g., LiOH; and hydroxyl groups formed from the reaction of broken bonds with water as in radiation damaged minerals. There is no evidence for the presence of the oxonium ion, H3O+, in common minerals, and the existing evidence for the occurrence of molecular hydrogen may better be explained by the presence of water or hydroxyl groups.</p