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A generalization of the concept of distance based on the simplex inequality Kiss, Gergely ; Marichal, Jean-Luc ; Teheux, Bruno in Beitraege zur Algebra und Geometrie = Contributions to Algebra and Geometry (2018), 59(2), 247266 We introduce and discuss the concept of \emph{$n$-distance}, a generalization to $n$ elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex ... [more ▼] We introduce and discuss the concept of \emph{$n$-distance}, a generalization to $n$ elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality \[ d(x_1, \ldots, x_n)~\leq~K\, \sum_{i=1}^n d(x_1, \ldots, x_n)_i^z{\,}, \qquad x_1, \ldots, x_n, z \in X, \] where $K=1$. Here $d(x_1,\ldots,x_n)_i^z$ is obtained from the function $d(x_1,\ldots,x_n)$ by setting its $i$th variable to $z$. We provide several examples of $n$-distances, and for each of them we investigate the infimum of the set of real numbers $K\in\left]0,1\right]$ for which the inequality above holds. We also introduce a generalization of the concept of $n$-distance obtained by replacing in the simplex inequality the sum function with an arbitrary symmetric function. [less ▲] Detailed reference viewed: 220 (31 UL)Pivotal decomposition schemes inducing clones of operations ; Teheux, Bruno in Beitraege zur Algebra und Geometrie = Contributions to Algebra and Geometry (2018), 59(1), 25-40 We study pivotal decomposition schemes and investigate classes of pivotally decomposable operations. We provide sufficient conditions on pivotal operations that guarantee that the corresponding classes of ... [more ▼] We study pivotal decomposition schemes and investigate classes of pivotally decomposable operations. We provide sufficient conditions on pivotal operations that guarantee that the corresponding classes of pivotally decomposable operations are clones, and show that under certain assumptions these conditions are also necessary. In the latter case, the pivotal operation together with the constant operations generate the corresponding clone. [less ▲] Detailed reference viewed: 122 (22 UL) |
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