The Brooks-Jewett theorem for \(k\)-triangular functions (Q2777506)

Let \(L\) be an orthomodular poset (OMP). A function \(\psi :L \rightarrow [0, \infty)\) for \(k\leq 1\) is a \(k\)-triangular function [\textit{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 \(a \bot b.\) NEWLINENEWLINENEWLINEThe 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 \textit{E. Pap}, loc. cit.] and a modification of \textit{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)]. NEWLINENEWLINENEWLINETheorem 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 \textit{E. Pap} [``A generalization of the diagonal theorem on a block-matrix'', Mat. Vesn., N. Ser. 11(26), 66-71 (1974; Zbl 0321.22001)].