The Lattice of Varieties of Completely Regular Semigroups

Polák’s theorem on the structure of the (lattice of) varieties of completely regular semigroups provides an isomorphic copy of the interval $[{\cal S,CR}]$ of varieties which contain semilattices in terms of certain functions. We give a variant of this theorem for the lattice ${\cal L(CR)}$ of all varieties of completely regular semigroups in terms of pairs with componentwise inclusion. The first entry of these pairs is a band variety and the second consists of a ?₀-tuple of members of ${\cal K}_0$ . Here ${\cal K}_0$ is the set of varieties which satisfy ${\cal V}_K={\cal V}$ where ${\cal V}_K$ is the least element of the K-class containing ${\cal V}$ . We have based the proof of our theorem on Polák’s theorem for the sake of expediency and comparison. It utilizes a set of varieties which we term canonical. Several corollaries treat various special cases.     

Ladders and canonical varieties of completely regular semigroups

A completely regular semigroup is a (disjoint) union of its (maximal) subgroups. We consider it here with the unary operation of inversion within its maximal subgroups. Their totality \(\mathcal {C}\mathcal {R}\) forms a variety whose lattice of subvarieties is denoted by \(\mathcal {L}(\mathcal {C}\mathcal {R})\). On it, one defines the relations \(\mathbf {B}^\wedge \) and \(\mathbf {B}^\vee \) by

$$\begin{aligned} \begin{array}{lll} \mathcal {U}\ \mathbf {B}^\wedge \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\cap \mathcal {B} =\mathcal {V}\cap \mathcal {B}, \\ \mathcal {U}\ \mathbf {B}^\vee \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\vee \mathcal {B} =\mathcal {V}\vee \mathcal {B} , \end{array} \end{aligned}$$

respectively, where \(\mathcal {B}\) denotes the variety of all bands. This is a study of the interplay between the \(\cap \)-subsemilatice \(\triangle \) of \(\mathcal {L}(\mathcal {C}\mathcal {R})\) of upper ends of \(\mathbf {B}^\wedge \)-classes and their \(\mathbf {B}^\vee \)-classes. The main tool is the concept of a ladder and their \(\mathbf {B}^\vee \)-classes, an indispensable part of the important Polák’s theorem providing a construction for the join of varieties of completely regular semigroups. The paper includes the tables of ladders of the upper ends of most \(\mathbf {B}^\wedge \)-classes. Canonical varieties consist of two ascending countably infinite chains which generate most of the upper ends of \(\mathbf {B}^\wedge \)-classes.

     

Normal cryptogroups with an associate subgroup

Let S be a semigroup. For a, x ∈ S such that a = axa, we say that x is an associate of a. A subgroup G of S which contains exactly one associate of each element of S is called an associate subgroup of S. It induces a unary operation in an obvious way, and we speak of a unary semigroup satisfying three simple axioms. A normal cryptogroup S is a completely regular semigroup whose H -relation is a congruence and S/H is a normal band. Using the representation of S as a strong semilattice of Rees matrix semigroups, in a previous communication we characterized those that have an associate subgroup. In this paper, we use that result to find three more representations of this semigroup. The main one has a form akin to the one of semigroups in which the identity element of the associate subgroup is medial.     

Binary relations as lattice isomorphisms

Summary In 1963, Zaretskiį established a one-to-one correspondence between the setB _X of binary relations on a set X and the set of triples of the form (W, ϕ, V) where W and V are certain lattices and ϕ: W→V is an isomorphism. We provide a multiplication for these triples making the Zaretskiį correspondence a semigroup isomorphism. In addition, we consider faithful representations ofB _X by pairs of partial transformations and also as the translational hull of its rectangular relations. Using these triples, we study idempotents, regular and completely regular elements and relationsH-equivalent to some relations with familiar properties such as reflexivity, transitivity, etc. Entrata in Redazione il 14 aprile 1998.     

Photophysics of Hypericin and Hypocrellin A in Complex with Subcellular Components: Interactions with Human Serum Albumin

Time-resolved fluorescence and absorption measurements are performed on hypericin complexed with human serum albumin, HSA (1:4, 1:1 and approximately 5:1 hypericin: HSA complexes). Detailed comparisons with hypocrellin A/HSA complexes (1:4 and 1:1) are made. Our results are consistent with the conclusions of previous studies indicating that hypericin binds to HSA by means of a specific hydrogen-bonded interaction between its carbonyl oxygen and the N1-H of the tryptophan residue in the IIA subdomain of HSA. (They also indicate that some hypericin binds nonspecifically to the surface of the protein.) A single-exponential rotational diffusion time of 31 ns is measured for hypericin bound to HSA, indicating that it is very rigidly held. Energy transfer from the tryptophan residue of HSA to hypericin is very efficient and is characterized by a critical distance of 94 A, from which we estimate a time constant for energy transfer of approximately 3 x 10(-15) s. Although it is tightly bound to HSA, hypericin is still capable of executing excited-state intramolecular proton (or hydrogen atom) transfer in the approximately 5:1 complex, albeit to a lesser extent than when it is free in solution. It appears that the proton transfer process is completely impeded in the 1:1 complex. The implications of these results for hypericin (and hypocrellin A) are discussed in terms of the mechanism of intramolecular excited-state proton transfer, the mode of binding to HSA and the light-induced antiviral and antitumor activity.     

A Comparison of the Fluorescence Spectra of Murine and Bovine Central Nervous System and Other Tissues

We describe a comparison of the fluorescence spectra of bovine tissues with murine tissues in order to determine whether spectral features are conserved and whether an appropriate and practical laboratory small animal model system could be identified to be used for investigation of tissue- and age-related fluorescence signal patterns. Recently it has been shown that spectral signatures of lipofuscin have enabled the detection of bovine central nervous system (CNS) tissue in meat products with high sensitivity (Schönenbrücher, H., Adhikary, R., Mukherjee, P., Casey, T.A., Rasmussen, M.A., Maistrovich, F.D., Hamir, A.N., Kehrli, M.J., Richt, J., Petrich, J.W. [2008] J Agric Food Chem 56 , 6220–6226). We report that brain and spinal cord of mice provide fluorescence spectra similar to those of bovine brain and spinal cord. It is concluded that murine CNS tissue is an appropriate model system for bovine CNS tissue for the development of fluorometric CNS detection assays.     

The superprism effect using large area 2D-periodic photonic crystal slabs

The “superprism effect” is an effect observed in photonic crystal structures whereby the direction of light propagation is extremely sensitive to the wavelength and angle of incidence. To realize the superprism effect, new structures are presented which rely on the sensitivity of the phase velocity in a two-dimensional (2D) photonic crystal slab to observe angular magnification outside the photonic crystal medium. Constant frequency contour calculations for a photonic crystal slab of finite thickness are used to predict the phase velocity superprism effect. Further analysis using 2D finite-difference time-domain simulations indicate that a large area photonic crystal and wide excitation beam are necessary for clear observation of the superprism effect. A fabrication technique is demonstrated to achieve the structure's required nanometer-sized features over centimeter-scale areas.     

Certain Partial Orders on Semigroups

Relations introduced by Conrad, Drazin, Hartwig, Mitsch and Nambooripad are discussed on general, regular, completely semisimple and completely regular semigroups. Special properties of these relations as well as possible coincidence of some of them are investigated in some detail. The properties considered are mainly those of being a partial order or compatibility with multiplication. Coincidences of some of these relations are studied mainly on regular and completely regular semigroups.     

Orders in Strict Regular Semigroups

 A subsemigroup S of a semigroup Q is an order in Q if for every there exist such that , where a and d are contained in (maximal) subgroups of Q, and and are their inverses in these subgroups. A regular semigroup S is strict if it is a subdirect product of completely (0-)simple semigroups. We construct all orders and involutions in Auinger’s model of a strict regular semigroup. This is used to find necessary and sufficient conditions on an involution on an order S in a strict regular semigroup Q for extendibility to an involution on Q. (Received 27 April 1999; in revised form 20 October 1999)