Abstract:
The aim of this note is to give a direct proof for the following result proved by Fountain and Lewin: {\em Let $\alg$ be an independence algebra of finite rank and let $a$ be a singular endomorphism of $\alg $. Then $a=e_1... e_n$ where $e^2_i=e_i$ and $rank(a)=rank(e_i)$.}

Abstract:
We provide short and direct proofs for some classical theorems proved by Howie, Levi and McFadden concerning idempotent generated semigroups of transformations on a finite set.

Abstract:
It is unthinkable to discuss learning without thinking of the merited prominence of reading and writing, since these are the essential actions for the production of different forms of systematized knowledge, among which is scientific knowledge. From this perspective, the present work proposes answering the following question: to what extent can the proposed pedagogy called reciprocal teaching aid in the development of the abilities of reading and interpretation of scientific texts? The discussion of a text found in a biology textbook was organized around four moments: questioning, summarizing, predicting and clarifying. The analysis of the data shows that the collective work helped in the comprehension of the text and, especially, in the understanding of the meaning of some scientific terms and permitted a deeper and more complex analysis of the text. This would hardly occur in a polarized reading situation in which the professor would present the expected interpretation.

Abstract:
this article attempts to systematize some contributions from contemporary theory (schumpeter, riker, mcgann, przeworski, cox, mccubbins, powell, crisp, arato and lijphart, among others) on the limits of preference aggregation and on the most propitious institutional bases for the development of accountability. through this analysis, we seek to demonstrate the link between minimalist theories of democracy and their normative bases, and in particular, the value that is placed on the dimension of accountability. it is our hypothesis that there is no absence of normativity in the minimalist-proceduralist-schumpeterian conception of democracy. we have tried to show this through analysis of the relationship between institutional design and accountability. we seek to demonstrate how the institutional design of democratic regimes influences the formation of the characteristics of a democracy, which can be understood as indicators of the quality of democracy (in other words, the normative dimension of democracy.) our analysis unfolds through the differentiation of majoritarian and proportionalist designs and the institutional and normative characteristics that are tied to them, attempting to make contemporary arguments on the impossibility of preference aggregation and the relationship between institutional design-accountability explicit. the methodology we use consists of systematizing central arguments and carrying out theoretical analysis of the books and articles of selected authors.

Abstract:
Brazil is presently going through its worse electricity supply crisis in fifty years. This happens after seven year efforts of market-oriented reforms, and inevitably raises the issue of whether the design and rhythm of the reform have been correct. The roots of the present crisis lie in a long period of underinvestment dating from the eighties; sector reforms were aimed at correcting this situation, but have been unsuccessful thus far. This article discusses the causes of this failure and attempts a way out of the present problems. The present crisis requires an emergency answer, but also a long term policy. I argue that such a policy must be based upon the acknowledgement that electricity demand in Brazil will tend to grow fast for the foreseeable future and that sector reform must be based upon dynamic rather than static efficiency. Furthermore, the large Brazilian hydropower system requires special treatment if we are to have investment in hydro and in thermal plants.

Abstract:
Let T(X) be the semigroup of full transformations on a finite set X with n elements. We prove that every subsemilattice of T(X) has at most 2^{n-1} elements and that there are precisely n subsemilattices of size exactly 2^{n-1}, each isomorphic to the semilattice of idempotents of the symmetric inverse semigroup on a set with n-1 elements.

Abstract:
Let $\Omega$ be a set of cardinality $n$, $G$ a permutation group on $\Omega$, and $f:\Omega\to\Omega$ a map which is not a permutation. We say that $G$ synchronizes $f$ if the semigroup $\langle G,f\rangle$ contains a constant map. The first author has conjectured that a primitive group synchronizes any map whose kernel is non-uniform. Rystsov proved one instance of this conjecture, namely, degree $n$ primitive groups synchronize maps of rank $n-1$ (thus, maps with kernel type $(2,1,\ldots,1)$). We prove some extensions of Rystsov's result, including this: a primitive group synchronizes every map whose kernel type is $(k,1,\ldots,1)$. Incidentally this result provides a new characterization of imprimitive groups. We also prove that the conjecture above holds for maps of extreme ranks, that is, ranks 3, 4 and $n-2$. These proofs use a graph-theoretic technique due to the second author: a transformation semigroup fails to contain a constant map if and only if it is contained in the endomorphism semigroup of a non-null (simple undircted) graph. The paper finishes with a number of open problems, whose solutions will certainly require very delicate graph theoretical considerations.

Abstract:
Let $X$ be a finite set such that $|X|=n$, and let $k< n/2$. A group is $k$-homogeneous if it has only one orbit on the sets of size $k$. The aim of this paper is to prove some general results on permutation groups and then apply them to transformation semigroups. On groups we find the minimum number of permutations needed to generate $k$-homogeneous groups (for $k\ge 1$); in particular we show that $2$-homogeneous groups are $2$-generated. We also describe the orbits of $k$-homogenous groups on partitions with $n-k$ parts, classify the $3$-homogeneous groups $G$ whose orbits on $(n-3)$-partitions are invariant under the normalizer of $G$ in $S_n$, and describe the normalizers of $2$-homogeneous groups in the symmetric group. Then these results are applied to extract information about transformation semigroups with given group of units, namely to prove results on their automorphisms and on the minimum number of generators. The paper finishes with some problems on permutation groups, transformation semigroups and computational algebra.

Abstract:
Let $X$ be a finite set such that $|X|=n$ and let $i\leq j \leq n$. A group $G\leq \sym$ is said to be $(i,j)$-homogeneous if for every $I,J\subseteq X$, such that $|I|=i$ and $|J|=j$, there exists $g\in G$ such that $Ig\subseteq J$. (Clearly $(i,i)$-homogeneity is $i$-homogeneity in the usual sense.) A group $G\leq \sym$ is said to have the $k$-universal transversal property if given any set $I\subseteq X$ (with $|I|=k$) and any partition $P$ of $X$ into $k$ blocks, there exists $g\in G$ such that $Ig$ is a section for $P$. (That is, the orbit of each $k$-subset of $X$ contains a section for each $k$-partition of $X$.) In this paper we classify the groups with the $k$-universal transversal property (with the exception of two classes of 2-homogeneous groups) and the $(k-1,k)$-homogeneous groups (for $2