Restriction semigroups and $$\lambda $$ λ -Zappa-Szép products

As generalizations of inverse semigroups, restriction semigroups and regular \(*\)-semigroups are investigated by many authors extensively in the literature. In particular, Lawson and Hollings have proved that the category of restriction semigroups together with prehomomorphisms (resp. (2,1,1)-homomorphisms) is isomorphic to the category of inductive categories together with ordered functors (resp. strongly ordered functors), which generalizes the well-known Ehresmann–Schein–Nambooripad theorem (ESN theorem for short) for inverse semigroups. On the other hand, Imaoka and Fujiwara have also obtained an ESN-type theorem for locally inverse regular \(*\)-semigroups. Recently, Jones generalized restriction semigroups and regular \(*\)-semigroups to P-restriction semigroups from a varietal perspective and considered the constructions of P-restriction semigroups by using Munn’s approach. In this paper, we shall study the class of P-restriction semigroups by using “category approach.” We introduce the notion of inductive generalized categories over local semilattices by which a class of P-restriction semigroups called locally restriction P-restriction semigroups is described. Moreover, we show that the category of locally restriction P-restriction semigroups together with (2,1,1)-prehomomorphisms (resp. (2,1,1)-homomorphisms) is isomorphic to the category of inductive generalized categories over local semilattices together with preadmissible mappings (resp. admissible mappings). Our work may be regarded as extending the ESN-type theorems for restriction semigroups and locally inverse regular \(*\)-semigroups, respectively.

The internal Zappa-Szep products emerge when a semigroup has the property that every element has a unique decomposition as a product of elements from two given subsemigroups. The external version constructed from actions of two semigroups on one another satisfying axiom derived by G. Zappa. We illustrate the correspondence between the two versions internal and the external of Zappa-Szep products of semigroups. We consider the structure of the internal Zappa-Szep product as an enlargement. We show how rectangular band can be described as the Zappa-Szep product of a left-zero semigroup and a right-zero semigroup. We find necessary and sufficient conditions for the Zappa-Szep product of regular semigroups to again be regular, and necessary conditions for the Zappa-Szep product of inverse semigroups to again be inverse. We generalize the Billhardt λ-semidirect product to the Zappa-Szep product of a semilattice E and a group G by constructing an inductive groupoid.

Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain. This paper is the sequel to two recent papers by the author, melding the results of the first, on membership in the variety $\mathbf{B}$ of left restriction semigroups generated by Brandt semigroups and monoids, with the connection established in the second between subvarieties of the variety $\mathbf{B}_{R}$ of two-sided restriction semigroups similarly generated and varieties of categories, in the sense of Tilson. We show that the respective lattices ${\mathcal{L}}(\mathbf{B})$ and ${\mathcal{L}}(\mathbf{B}_{R})$ of subvarieties are almost isomorphic, in a very specific sense. With the exception of the members of the interval $[\mathbf{D},\mathbf{D}\vee \mathbf{M}]$, every subvariety of $\mathbf{B}$ is induced from a member of $\mathbf{B}_{R}$ and vice versa. Here $\mathbf{D}$ is generated by the three-element left restriction semigroup $D$ and $\mathbf{M}$ is the variety of monoids. The analogues hold for pseudovarieties.

Partial actions and proper extensions of two-sided restriction semigroups

Rees Matrix Covers and Semidirect Products of Regular Semigroups

We consider semigroups such that the universal left congruence omega ^{ell } is finitely generated. Certainly a left noetherian semigroup, that is, one in which all left congruences are finitely generated, satisfies our condition. In the case of a monoid the condition that omega ^{ell } is finitely generated is equivalent to a number of pre-existing notions. In particular, a monoid satisfies the homological finiteness condition of being of type left-{text {FP}}_1 exactly when omega ^{ell } is finitely generated.Our investigations enable us to classify those semigroups such that omega ^{ell } is finitely generated that lie in certain important classes, such as strong semilattices of semigroups, inverse semigroups, Rees matrix semigroups (over semigroups) and completely regular semigroups. We consider closure properties for the class of semigroups such that omega ^{ell } is finitely generated, including under morphic image, direct product, semi-direct product, free product and 0-direct union. Our work was inspired by the stronger condition, stated for monoids in the work of White, of being pseudo-finite. Where appropriate, we specialise our investigations to pseudo-finite semigroups and monoids. In particular, we answer a question of Dales and White concerning the nature of pseudo-finite monoids.

We compare the two recently introduced semidirect product operations * r and * rr within the lattice of e-varieties of locally inverse semigroups. For each e-variety $$ \mathcal{V}$$ which contains all rectangular bands and is properly contained in the e-variety of all completely simple semigroups, the inclusions $$ \mathcal{S} *_{rr} \mathcal{V} \subsetneq \mathcal {S} *_r \mathcal{V} \subsetneq \mathcal{S} *_{rr} (\mathcal{A}_q \circ \mathcal{V}) $$ are proved where $$ \mathcal{S} $$ is the e-variety of all semilattices and $$ \mathcal{A}_q $$ the variety of all abelian groups of exponent dividing q where q is any integer greater than one. Some consequences for the class of finite locally inverse semigroups are also obtained.

We examine a certain embedding problem for groups that have a presentation described by partial automorphisms. Semidirect products and HNN extensions have such a presentation. The embedding problem is closely related to systems of partial automorphisms, which are formalized by the concept of inverse semigroups. A necessary and sufficient condition for a group to be embedded in a certain sense is obtained by geometric methods using van Kampen diagrams.

We study automaton structures, i.e., groups, monoids and semigroups generated by an automaton, which, in this context, means a deterministic finite-state letter-to-letter transducer. Instead of considering only complete automata, we specifically investigate semigroups generated by partial automata. First, we show that the class of semigroups generated by partial automata coincides with the class of semigroups generated by complete automata if and only if the latter class is closed under removing a previously adjoined zero, which is an open problem in (complete) automaton semigroup theory stated by Cain. Then, we show that no semidirect product (and, thus, also no direct product) of an arbitrary semigroup with a (non-trivial) subsemigroup of the free monogenic semigroup is an automaton semigroup. Finally, we concentrate on inverse semigroups generated by invertible but partial automata, which we call automaton-inverse semigroups, and show that any inverse automaton semigroup can be generated by such an automaton (showing that automaton-inverse semigroups and inverse automaton semigroups coincide).

Analysis of volume datasets often involves peering inside the volume to understand internal structures. Traditional approaches involve removing part of the volume through slicing, but this can result in the loss of context. Focus+context visualization techniques can distort part of the volume, or can assume prior definition of a region of interest or segmentation of layers of the volume. We propose a new bimanual 3D interaction technique, called Volume Cracker (VC), which allows the user to crack open a raw volume like a book to analyze the internal structures. VC preserves context by always displaying all the voxels, and by connecting the sub-volumes with curves joining the cracked faces. We discuss the design choices that we made, based on observations from prior user studies, input from domain scientists, and design studios. We also report the results of a user study comparing VC with a standard desktop interaction technique and a standard 3D bimanual interaction technique. The study used tasks from two categories of a generic volume analysis task taxonomy. We found VC had significant advantages over the other two techniques for search and pattern recognition tasks.

<p>Catastrophic coastal flooding is one of the main forcing agents of short-term coastal system changes and represents a major threat to human activities concentrated along the coasts worldwide, particularly in the light of ongoing climate change. In order to better understand the frequency and character of catastrophic marine inundation events in the past as well as to predict future trends the knowledge on the long-time records of Holocene coastal flooding chronologies is necessary.</p><p>The southern coast of the Baltic Sea is an important study area because it is exposed to both, (north) westerly and (north) easterly storms and corresponding seiche effects. Moreover, the negligible tidal forcing does not bias the elevated water table of marine water surge events, so the true coastal flooding signal is preserved in the sedimentary record (Hippensteel, 2010). Furthermore, as demonstrated in a recent study by Piotrowski et al. (2017) in the area of Polish coast the low lying marsh areas behind coastal dunes or at river mouths are promising sedimentary environments to provide with record of catastrophic coastal flooding.</p><p>The poster reviews the most up-to-date state of palaeo-tempestological research within the southern Baltic Sea coast summing up the newest findings of the CatFlood project launched in March 2019. The overview of topographical and geomorphological characteristics of field locations, which are most prone to marine coastal flooding and preservation of sedimentological evidence for these catastrophic events will be given based on the pilot study within 16 field locations. The study sites are scattered along whole Polish Baltic Sea coast. Furthermore, in depth observations of features of deposits associated with marine inundation events is provided based on the detailed analysis of sediments from four selected key field locations. The event- layers characteristics are reconstructed by standard techniques such as grain size, shape and texture, heavy mineral composition, mineral versus organic matter ratio analyses. Above that the analysis of internal structure of flooding deposits in microscale is described from thin sections. The composite chronologies and the high resolution age control based on both <sup>14</sup>C dating and <sup>210</sup>Pb/<sup>137</sup>Cs provides with insights into the chronology of these events. A new approach is the application of seda-DNA analysis in deciphering the marine character of event deposits.</p><p><strong>

Direct STM observation of native biomolecules has proven feasible. In this study we have used STM to image two filamentous protein components of the intracellular cytoskeleton: microtubules (MT) and intermediate filaments (IF). MT are 25 nm diameter cylinders comprised of 13 ‘‘protofilaments’’ which are linear chains of 8 nm × 4 nm × 4 nm ‘‘dimer’’ subunits. More variable than MT, IF are 10 nm diameter coils comprised of from 4 to 8 ‘‘subfilaments’’ which are chains of ‘‘tetramer’’ subunits. MT were isolated from porcine brain and IF from cell culture by standard techniques. Preparation/stabilization factors included magnesium buffer, 0.8 molar glycerol, and 0.1% glutaraldehyde. Samples were scanned on graphite in a Nanoscope I or II STM (Digital Instruments, Santa Barbara, California). STM images of MT demonstrated flattened 25 nm diameter structures composed of 4 nm wide protofilaments. Processed inverted images showed rows of 8 nm × 4 nm subunits. STM images of IF showed flattened, parallel 10 nm filaments comprised of coiled chains of 6–12 nm tetramer subunits. Thus STM demonstrated internal structure of two different classes of cytoskeletal proteins. STM and related techniques have an important future in biomolecular studies.

Volcanic eruptions necessitate precise monitoring of magma pressure and inflation for improved forecasting. Understanding deep magma storage is crucial for hazard assessment, yet imaging these systems is challenging due to complex heterogeneities that disrupt standard seismic migration techniques. Here we map the magmatic and hydrothermal system of the La Soufrière volcano in Guadeloupe by analyzing seismic noise data from a sparse geophone array under a matrix formalism. Seismic noise interferometry provides a reflection matrix containing the signature of echoes from deep heterogeneities. Using wave correlations resistant to disorder, matrix imaging successfully unscrambles wave distortions, revealing La Soufrière’s internal structure down to 10 km with 100 m resolution. This method surpasses the diffraction limit imposed by geophone array aperture, providing crucial data for modeling and high-resolution monitoring. We see matrix imaging as a revolutionary tool for understanding volcanic systems and enhancing observatories’ abilities to monitor dynamics and forecast eruptions.

Proper extensions that are “injective on ℒ-related idempotents” of ℛ-unipotent semigroups, and much more generally of the class of generalised left restriction semigroups possessing the ample and congruence conditions, referred to here as glrac semigroups, are described as certain subalgebras of a λ-semidirect product of a left regular band by an ℛ-unipotent or by a glrac semigroup, respectively. An example of such is the generalized Szendrei expansion.

Geoid anomalies offer crucial information on the internal density structure of the Earth, and thus, on its constitution and dynamic state. In order to interpret geoid undulations in terms of depth, magnitude and lateral extension of density anomalies in the lithosphere and upper mantle, the effects of lower mantle density anomalies need to be removed from the full geoid (thus obtaining the residual “upper mantle geoid”). However, how to achieve this seemingly simple filtering exercise has eluded consensus for decades in the solid Earth community. While there is wide agreement regarding the causative masses of degrees >10 in spherical harmonic expansions of the upper mantle geoid, those contributing to degrees <7–8 remain ambiguous. Here we use spherical harmonic analysis and recent tomography and density models from joint seismic‐geodynamic inversions to derive a representative upper mantle geoid, including the contributions from low harmonic degrees. We show that the upper mantle geoid contains important contributions from degrees 5 and 6 and interpret the causative masses as arising from the coupling between the long‐wavelength lithospheric structure and the sublithospheric upper mantle convection pattern. Importantly, the contributions from degrees 3 < l < 8 do not show a simple power‐law behavior (e.g., Kaula's rule), which precludes the use of standard filtering techniques in the spectral domain. Our upper mantle geoid model will be useful in studies of (a) lithospheric structure, (b) dynamic topography and mantle viscosity, (c) lithosphere‐asthenosphere interactions and (d) the global stress field within the lithosphere and its associated hazards.