Several authors have defined operational quantities derived from the norm of an operator between Banach spaces. This situation is generalized in this paper and we present a general framework in which we derivate several maps from an initial one , where is a set endowed with two orders, and , related by certain conditions. We obtain only three different derivated maps, if the initial map is bounded and monotone.
The paper contains characterizations of generators and cyclic projective generators in the category of ordered right acts over an ordered monoid.
We proved in an earlier work that any existence variety of regular algebras is generated by its simple unital Artinian members, while any existence variety of Arguesian sectionally complemented lattices is generated by its simple members of finite length. A characterization of the class of simple unital Artinian members [members of finite length, respectively] of such varieties is given in the present paper.
Dans la deuxième moitié du xixe siècle, une ambition commune anime le groupe de mathématiciens dont les travaux sont présentés ici : contribuer à la diffusion de l’esprit scientifique auprès d’un large public. Le lieu d’expression de ce groupe est l’Association française pour l’avancement des sciences, créée en 1872, après la défaite de la France au cours du conflit franco-prussien. Rendre la science populaire, tel est le but poursuivi. Afin de répondre à cet objectif, les questions mathématiques...
In this paper we first study what changes occur in the posets of irreducible elements when one goes from an arbitrary Moore family (respectively, a convex geometry) to one of its lower covers in the lattice of all Moore families (respectively, in the semilattice of all convex geometries) defined on a finite set. Then we study the set of all convex geometries which have the same poset of join-irreducible elements. We show that this set—ordered by set inclusion—is a ranked join-semilattice and we...
The concept of a Goldie extending module is generalized to a Goldie extending element in a lattice. An element of a lattice with is said to be a Goldie extending element if and only if for every there exists a direct summand of such that is essential in both and . Some properties of such elements are obtained in the context of modular lattices. We give a necessary condition for the direct sum of Goldie extending elements to be Goldie extending. Some characterizations of a decomposition...
We study the integral quaternions and the integral octonions along the combinatorics of the -cell, a uniform polytope with the symmetry , and the Gosset polytope with the symmetry . We identify the set of the unit integral octonions or quaternions as a Gosset polytope or a -cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the or actions on the or the -cell, respectively. Moreover, we show that each...
A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system. Degenerations of such orbits induce a natural partial order on the set of rook placements. We study combinatorial structure of the set of rook placements in with respect to a slightly different order and prove that this poset is...
In this paper we apply the notion of cell of a lattice for dealing with graph automorphisms of lattices (in connection with a problem proposed by G. Birkhoff).