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Summary: In this paper we consider the monoid $\mathcal {OR}_{n}$ of all full transformations on a chain with $n$ elements that preserve or reverse the orientation, as well as its submonoids $\mathcal {OD}_{n}$ of all order-preserving or order-reversing elements, $\mathcal {OP}_{n}$ of all orientation-preserving elements and $\mathcal {O}_{n}$ of all order-preserving elements. By making use of some well known presentations, we show that each of these four monoids is a quotient of a bilateral semidirect product of two of its remarkable submonoids.

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The elastodynamic response of saturated poroelastic media is modelled approximating independently the solid and seepage displacements in the domain and the force and pressure components on the boundary of the element. The domain and boundary approximation bases are used to enforce on average the dynamic equilibrium and the displacement continuity conditions, respectively. The resulting solving system is Hermitian, except for the damping term, and its coefficients are defined by boundary integral expressions as a Trefftz basis is used to set up the domain approximation. This basis is taken from the solution set of the governing differential equation and models the free-field elastodynamic response of the medium. This option justifies the relatively high levels of performance that are illustrated with the time domain analysis of unbounded domains.

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