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Blocks of pseudo-effect algebras with the Riesz interpolation property. (English)
Soft Comput. 7, No.7, 441-445 (2003).
Pseudo-effect algebras are a noncommutative counterpart of effect algebras [see {\it A. Dvurečenskij} and {\it T. Vetterlein}, Int. J. Theor. Phys. 40, 685-701 (2001; Zbl 0994.81008) for more details]. A pseudo-effect algebra $E$ is said to have the Riesz Interpolation Property (RIP) if the inequalities $x_i \le y_j$ with $1 \le i,j \le 2$ hold in $E$ only if there is an element $z \in E$ such that $x_i \le z \le y_j$ for $1 \le i,j \le 2$. It is shown that the partial join and meet operations in a pseudo-effect algebra with RIP have a number of useful algebraic properties. {\it Z. Riečanová} proved [Int. J. Theor. Phys. 39, 231-237 (2000; Zbl 0968.81003)] that a lattice-ordered effect algebra can be covered by maximal subsets of mutually compatible elements, which all are MV-algebras. Similar results for effect algebras with RIP and for lattice-ordered pseudo-effect algebras were recently obtained by the author [Int. J. Theor. Phys. 41, 221-229 (2002; Zbl 1022.06005)] and {\it A. Dvurečenskij} and {\it T. Vetterlein} [Demonstr. Math. 36, 261-282 (2003; Zbl 1039.03049)], respectively. The main theorem of the paper under review extends these results to pseudo-effect algebras with RIP having, in addition, one more property concerning compatibility of elements; all effect algebras and pseudo MV-algebras have this additional property.
Reviewer: Jānis Cīrulis (Riga)