Free poset algebras and combinatorics of cones (Q2385435)

The concept of the length of an element was introduced by \textit{M. Pouzet} and \textit{I. Rival} [Algebra Univers. 18, 295--307 (1984; Zbl 0545.06005)]. An explicit formula in the case of interval algebras was presented by \textit{H. Si Kaddour} [ibid. 44, 195--198 (2000; Zbl 1013.06015)] and generalized to pseudo-tree algebras by \textit{M. Bekkali} and \textit{D. Zhani} [Rev. Mat. Complut. 17, 169--179 (2004; Zbl 1050.06004)] and by \textit{D. Zhani} [N. Z. J. Math. 35, No. 2, 215--223 (2006)]. The main goal of this paper is to define the above notion in free poset algebras by using the support of elements in a very canonical manner. All the results of the paper are new, interesting and carefully proved, and so they contribute to the development of this research domain.