The reticulation of a hoop Cheptea
In this paper we study the hoops, ordered structures introduced by B. Bosbach under the name of complemented semigroups. We built a reticulation of a hoop, which is a bounded distributive lattice such that there exists a homeomorphism between the prime spectrum of the hoop and the prime spectrum of the lattice. We find an axiomatic definition of the reticulation and we prove the existence of it. We prove the unicity of the reticulation modulo a lattices isomorphism. We built a covariant functor between the category of bounded distributive lattices and the category of bounded hoops, which allows us to obtain new properties from the properties of lattices.
Reticulation of a 0-distributive Lattice
A congruence relation θ on a 0-distributive lattice is defined such that the quotient lattice L/θ is a distributive lattice and the prime spectrum of L and of L/θ are homeomorphic. Also it is proved that the minimal prime spectrum (maximal spectrum) of L is homeomorphic with the minimal prime spectrum (maximal spectrum) of L/θ.
Three-dimensional elasticity problems for the prismatic bar
A general solution is given to the three-dimensional linear elastic problem of a prismatic bar subjected to arbitrary tractions on its lateral surfaces, subject only to the restriction that they can be expanded as finite power series in the axial coordinate z. The solution is obtained by repeated differentiation of the tractions with respect to z, establishing a set of sub-problems . A recursive procedure is then developed for generating the solution to from that for . This procedure involves three steps: integration of the stress and displacement fields with respect to z, using an appropriate Papkovich–Neuber (P–N) representation; solution of two-dimensional in-plane and antiplane corrective problems for the tractions in that are independent of z; and expression of these corrective solutions in P–N form. The method is illustrated by an example.