Evolution algebras and graphs

A digraph is attached to any evolution algebra. This graph leads to some new purely algebraic results on this class of algebras and allows for some new natural proofs of known results. Nilpotency of an evolution algebra will be proved to be equivalent to the nonexistence of oriented cycles in the graph. Besides, the automorphism group of any evolution algebra $E$ with $E=E^2$ will be shown to be always finite.Comment: 10 page

On a class of baric algebras

AbstractIt is known that baric algebras satisfying the identity (x2)2 = w(x)x3 have idempotent elements and every linear form w: A → K is a multiplicative map. We prove that these algebras are Jordan-Bernstein of order 2 and special train algebras. Moreover, as a corollary we obtain that the train equation of these algebras is x4 − w(x)x3 = 0, and we give examples of baric algebras satisfying x4 − w(x)x3 = 0 but not satisfying (x2)2 = w(x)x3

On some Jordan baric algebras

Several classes of baric algebras studied by different authors will be given a unified treatment, using the technique of gametization introduced by Mallol et al. Many of these algebras will be shown to be either Jordan algebras or to be closely related to them.Comment: 11 page

Generalized Jordan algebras

AbstractWe study commutative algebras which are generalizations of Jordan algebras. The associator is defined as usual by (x,y,z) = (xy)z−x(yz). The Jordan identity is (x2,y,x) = 0. In the three generalizations given below, t, β, and γare scalars. ((xx)y)x+t((xx)x)y=0, ((xx)x)(yx)−(((xx)x)y)x=0, β((xx)y)x+γ((xx)x)y−(β+γ)((yx)x)x=0. We show that with the exception of a few values of the parameters, the first implies both the second and the third. The first is equivalent to the combination of ((xx)x)x=0 and the third. We give examples to show that our results are in some reasonable sense, the best possible

Solvability of Commutative Right-Nilalgebras Satisfying (b (aa)) a= b ((aa) a)

We study commutative right-nilalgebras of right-nilindex four satisfying the identity (b(aa))a = b((aa)a). Our main result is that these algebras are solvable and not necessarily nilpotent. Our results require characteristic ≠ 2, 3, 5

Effects of One Laptop Per Child Programs in the Daily Lives of People Excluded from the Community: The Role of South European Rural Schools

he aim of this study was to learn how the One Laptop per Child (OLPC) program in rural schools of a Southern European country offers opportunities to children, helps parents to get information and creates links with families and neighbours. The methodology was qualitative, with deep interviews with teachers, principals, parents and students from five Spanish rural schools. Among the findings, we observed that schools have new opportunities to link with communities and to communicate with parents when the curriculum is open and narrative. The shortcomings of policies, the academic role of projects and the resistance and prejudice of parents are preventing a better rapport between schools and communities in their fight against exclusio