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 GG-acts

The study of the free idempotent generated semigroup $\mathrm{IG}(E)$ over a biordered set $E$ 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 $\mathrm{IG}(E)$ in the case $E$ is the biordered set of a wreath product $G\wr \mathcal{T}_n$, where $G$ is a group and $\mathcal{T}_n$ is the full transformation monoid on $n$ elements. This wreath product is isomorphic to the endomorphism monoid of the free $G$-act $F_n(G)$ on $n$ generators, and this provides us with a convenient approach. We say that the rank of an element of $F_n(G)$ is the minimal number of (free) generators in its image. Let $\varepsilon=\varepsilon^2\in F_n(G).$ For rather straightforward reasons it is known that if $\mathrm{rank}\,\varepsilon =n-1$ (respectively, $n$), then the maximal subgroup of $\mathrm{IG}(E)$ containing $\varepsilon$ is free (respectively, trivial). We show that if $\mathrm{rank}\,\varepsilon =r$ where $1\leq r\leq n-2$, then the maximal subgroup of $\mathrm{IG}(E)$ containing $\varepsilon$ is isomorphic to that in $F_n(G)$ and hence to $G\wr \mathcal{S}_r$, where $\mathcal{S}_r$ is the symmetric group on $r$ elements. We have previously shown this result in the case $r=1$; however, for higher rank, a more sophisticated approach is needed. Our current proof subsumes the case $r=1$ and thus provides another approach to showing that any group occurs as the maximal subgroup of some $\mathrm{IG}(E)$. On the other hand, varying $r$ again and taking $G$ 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 $\mathcal{T}_n.$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