On finite monoids of cellular automata.

For any group G and set A, a cellular automaton over G and A is a transformation τ:AG→AGτ:AG→AG defined via a finite neighbourhood S⊆GS⊆G (called a memory set of ττ) and a local function μ:AS→Aμ:AS→A. In this paper, we assume that G and A are both finite and study various algebraic properties of the finite monoid CA(G,A)CA(G,A) consisting of all cellular automata over G and A. Let ICA(G;A)ICA(G;A) be the group of invertible cellular automata over G and A. In the first part, using information on the conjugacy classes of subgroups of G, we give a detailed description of the structure of ICA(G;A)ICA(G;A) in terms of direct and wreath products. In the second part, we study generating sets of CA(G;A)CA(G;A). In particular, we prove that CA(G,A)CA(G,A) cannot be generated by cellular automata with small memory set, and, when G is finite abelian, we determine the minimal size of a set V⊆CA(G;A)V⊆CA(G;A) such that CA(G;A)=⟨ICA(G;A)∪V⟩CA(G;A)=⟨ICA(G;A)∪V⟩

Restriction semigroups and λ -Zappa-Szép products

The aim of this paper is to study λ-semidirect and λ-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product S⋈ T where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-Schein-Nambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are λ-semidirect products and λ-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of λ-semidirect products

D-semigroups and constellations

In a result generalising the Ehresmann–Schein–Nambooripad Theorem relating inverse semigroups to inductive groupoids, Lawson has shown that Ehresmann semigroups correspond to certain types of ordered (small) categories he calls Ehresmann categories. An important special case of this is the correspondence between two-sided restriction semigroups and what Lawson calls inductive categories. Gould and Hollings obtained a one-sided version of this last result, by establishing a similar correspondence between left restriction semigroups and certain ordered partial algebras they call inductive constellations (a general constellation is a one-sided generalisation of a category). We put this one-sided correspondence into a rather broader setting, at its most general involving left congruence D-semigroups (which need not satisfy any semiadequacy condition) and what we call co-restriction constellations, a finitely axiomatized class of partial algebras. There are ordered and unordered versions of our results. Two special cases have particular interest. One is that the class of left Ehresmann semigroups (the natural one-sided versions of Lawson’s Ehresmann semigroups) corresponds to the class of co-restriction constellations satisfying a suitable semiadequacy condition. The other is that the class of ordered left Ehresmann semigroups (which generalise left restriction semigroups and for which semigroups of binary relations equipped with domain operation and the inclusion order are important examples) corresponds to a class of ordered constellations defined by a straightforward weakening of the inductive constellation axioms

Functional microarray analysis suggests repressed cell-cell signaling and cell survival-related modules inhibit progression of head and neck squamous cell carcinoma

<p>Abstract</p> <p>Background</p> <p>Cancer shows a great diversity in its clinical behavior which cannot be easily predicted using the currently available clinical or pathological markers. The identification of pathways associated with lymph node metastasis (N+) and recurrent head and neck squamous cell carcinoma (HNSCC) may increase our understanding of the complex biology of this disease.</p> <p>Methods</p> <p>Tumor samples were obtained from untreated HNSCC patients undergoing surgery. Patients were classified according to pathologic lymph node status (positive or negative) or tumor recurrence (recurrent or non-recurrent tumor) after treatment (surgery with neck dissection followed by radiotherapy). Using microarray gene expression, we screened tumor samples according to modules comprised by genes in the same pathway or functional category.</p> <p>Results</p> <p>The most frequent alterations were the repression of modules in negative lymph node (N0) and in non-recurrent tumors rather than induction of modules in N+ or in recurrent tumors. N0 tumors showed repression of modules that contain cell survival genes and in non-recurrent tumors cell-cell signaling and extracellular region modules were repressed.</p> <p>Conclusions</p> <p>The repression of modules that contain cell survival genes in N0 tumors reinforces the important role that apoptosis plays in the regulation of metastasis. In addition, because tumor samples used here were not microdissected, tumor gene expression data are represented together with the stroma, which may reveal signaling between the microenvironment and tumor cells. For instance, in non-recurrent tumors, extracellular region module was repressed, indicating that the stroma and tumor cells may have fewer interactions, which disable metastasis development. Finally, the genes highlighted in our analysis can be implicated in more than one pathway or characteristic, suggesting that therapeutic approaches to prevent tumor progression should target more than one gene or pathway, specially apoptosis and interactions between tumor cells and the stroma.</p

Semigroups with zero whose idempotents form a subsemigroup

The structure of a categorical, E*-dense, E*-unitary E-semigroup S is elucidated in terms of a 'B-quiver', where B is a primitive inverse semigroup. In the case where S is strongly categorical, B is a Brandt semigroup. A covering theorem is also proved, to the effect that every categorical E*-dense E-semigroup has a cover which is a categorical, E*-dense, E*-unitary E-semigroup.</p