295 research outputs found
Every group is the maximal subgroup of a naturally occurring free idempotent generated semigroup
Gray and Ruskuc have shown that any group G occurs as the maximal subgroup of
some free idempotent generated semigroup IG(E) on a biordered set of
idempotents E, thus resolving a long standing open question. Given the group G,
they make a careful choice for E and use a certain amount of well developed
machinery. Our aim here is to present a short and direct proof of the same
result, moreover by using a naturally occuring biordered set.
More specifically, for any free G-act F_n(G) of finite rank at least 3, we
have that G is a maximal subgroup of IG(E) where E is the biordered set of
idempotents of End F_n(G). Note that if G is finite then so is End F_n(G)
Axiomatisability problems for S-posets
Let C be a class of algebras of a given fixed type t. Associated with the
type is a first order language L_t. One can then ask the question, when is the
class C axiomatisable by sentences of L_t. In this paper we will be considering
axiomatisability problems for classes of left S-posets over a pomonoid S (that
is, a monoid S equipped with a partial order compatible with the binary
operation). We aim to determine the pomonoids S such that certain categorically
defined classes are axiomatisable.
The classes we consider are the free S-posets, the projective S-posets and
classes arising from flatness properties. Some of these cases have been studied
in a recent article by Pervukhin and Stepanova. We present some general
strategies to determine axiomatisability, from which their results for the
classes of weakly po-flat and po-flat S-posets will follow. We also consider a
number of classes not previously examined
Free idempotent generated semigroups and endomorphism monoids of free -acts
The study of the free idempotent generated semigroup over a
biordered set began with the seminal work of Nambooripad in the 1970s and
has seen a recent revival with a number of new approaches, both geometric and
combinatorial. Here we study in the case is the biordered
set of a wreath product , where is a group and
is the full transformation monoid on elements. This wreath
product is isomorphic to the endomorphism monoid of the free -act
on generators, and this provides us with a convenient approach.
We say that the rank of an element of is the minimal number of
(free) generators in its image. Let For
rather straightforward reasons it is known that if (respectively, ), then the maximal subgroup of
containing is free (respectively, trivial). We show that if
where , then the maximal
subgroup of containing is isomorphic to that in
and hence to , where is the
symmetric group on elements. We have previously shown this result in the
case ; however, for higher rank, a more sophisticated approach is needed.
Our current proof subsumes the case and thus provides another approach to
showing that any group occurs as the maximal subgroup of some .
On the other hand, varying again and taking to be trivial, we obtain an
alternative proof of the recent result of Gray and Ru\v{s}kuc for the biordered
set of idempotents of Comment: 35 page
Glycolytic Metabolism of Macrophages Differs by Spatial Location and Subset in Tuberculous Granulomas
Tuberculosis (TB), caused by infection with Mycobacterium tuberculosis (Mtb), triggers the formation of granulomas in the host. Granulomas are composed of many different types of host immune cells including T and B lymphocytes, macrophages, and neutrophils (PMN). The metabolic pathways used by immune cells in granulomas are important for cell function and glycolytic metabolic pathways in granulomas may be involved limiting or promoting disease. By understanding immunometabolism in TB granulomas, we can improve diagnosis and potentially create new therapeutics to combat TB disease. GLUT1 is a glucose transporter that transports glucose into the cell. To determine what cell subsets express GLUT1 in a granuloma, IHC was performed on lung granuloma-containing tissue sections from non-human primates (NHP) that were experimentally infected with Mtb. These slides were stained for macrophage markers including CD11c and CD163, and GLUT1, before being imaged by fluorescence microscopy. Image analysis was performed using ImageJ to determine the total pixel area and percent pixel area of each cell marker occupied in granuloma cross sections, as well as co-localization between the macrophage markers and GLUT1. To identify if there is a relationship between glucose uptake and mycobacterial antigens, we performed a glucose uptake assay and hypoxic experiments using 2-NBDG on monocyte-derived macrophages that were stimulated with inactivated Mtb. Image analysis revealed co-localization of different cell markers and GLUT1. When looking at co-localization of CD11c and CD163 with GLUT1, we found that CD11c+CD163- epithelioid macrophages, the cells in granulomas that are most commonly infected with Mtb, expressed more GLUT1 than interstitial and alveolar macrophages. Moreover, we see evidence that macrophages increase their glucose (2-NBDG) uptake when stimulated with inactivated Mtb. Identifying immune cells that express GLUT1 and what triggers these immune cells to switch to glycolytic metabolism will help us further understand overall granuloma metabolism and cell differentiation after Mtb infection. In terms of public health, by better understanding the immunometabolism of granulomas, it will improve monitoring of disease progression or to aid in the treatment of Mtb infection to help reduce tuberculosis cases worldwide
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