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The Brooks-Jewett theorem for $$k$$-triangular functions. (English) Zbl 0986.03048
Let $$L$$ be an orthomodular poset (OMP). A function $$\psi :L \rightarrow [0, \infty)$$ for $$k\leq 1$$ is a $$k$$-triangular function [E. Pap, Null-additive set functions, Kluwer, Dordrecht (1995; Zbl 0856.28001)] if it satisfies $$\psi(0) =0$$ and $$[\psi(a)-k \psi(b) \leq \psi(a \vee b) \leq \psi(a) + k \psi(b)]$$ for all $$a,b \in L$$ with $$a \bot b.$$
The main result is a generalization of the Brooks-Jewett convergence theorem (Theorem 6) for $$k$$-triangular functions defined on OMP $$L$$ with subsequential completeness property, i.e., for every orthogonal sequence in $$L$$ there exists a subsequence which has supremum in $$L.$$ The proof is based on a diagonal method (close to the sliding hump method) [see E. Pap, loc. cit.] and a modification of H. Weber [“Compactness in spaces of group-valued contents, the Vitali-Hahn-Saks theorem and Nikodým’s boundedness theorem”, Rocky Mt. J. Math. 16, 253-275 (1986; Zbl 0604.28006)].
Theorem 6 is extended to functions with values in a commutative semigroup with a neutral element $$0$$ endowed with a function $$f\: S \to [0, \infty)$$ satisfying $$f(0)=0$$ and $$[|f(x+y)-f(x) |\leq f(y)$$ for all $$x,y \in S],$$ see E. Pap [“A generalization of the diagonal theorem on a block-matrix”, Mat. Vesn., N. Ser. 11(26), 66-71 (1974; Zbl 0321.22001)].

##### MSC:
 03G12 Quantum logic 28A33 Spaces of measures, convergence of measures
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##### References:
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