In this paper we investigate the mathematical foundations of the notion of similarity between objects in relation to the granulations on a decision table $\mathfrak{D}$. First of all, we compare the endogenous granulation induced by Pawlak’s indiscernibility with the exogenous granulation induced by a similarity measure $\zeta$ defined on pairs of objects and assuming values in the unit interval. To this aim, the starting point of our analysis is the introduction of the notion of refinement of the granulation induced by an attribute subset A through the object similarity measure $\zeta$. More in detail, we say that $\zeta$ refines the granulation induced by A if $\zeta$ assumes value 1 on a pair of objects if and only if they are A-indiscernible. Next, starting from two given families $\rho$ and $\nu$ of numerical maps defined on pairs of admissible values of $\mathfrak{D}$, we determine a broad class of potential similarity measures on the objects of $\mathfrak{D}$ refining, sometimes under some specific additional hypotheses, the A-granulation on the object set of $\mathfrak{D}$. With regard to a such class of similarity measures, we establish several mathematical properties. Finally, we focus our attention to the analysis of specific pairs of numerical maps $\rho$ and $\nu$ that have been classically studied in literature and, for each of them, we exhibit the main properties with respect to the aforementioned refinement of granulation.

在本文中，我们研究了与决策表上的粒度相关的对象之间相似性概念的数学基础。 $\mathfrak{d}$。首先，我们将Pawlak的不可区分性引起的内生颗粒与相似性度量引起的外源性颗粒进行比较$\zeta$在对象对上定义并采用单位间隔中的值。为此，我们分析的出发点是通过对象相似性度量引入由属性子集A诱导的细化概念$\zeta$。更详细地说，我们说$\zeta$ 细化诱导造粒甲如果$\zeta$当且仅当它们是A不可区分时，才在一对对象上假定值1 。接下来，从两个给定的家庭开始$\rho$ 和 $\nu$ 在成对的允许值对上定义的数字图 $\mathfrak{d}$，我们针对的对象确定了一大类潜在的相似性度量 $\mathfrak{d}$在某些特定的附加假设下，细化对象集上的A-粒化$\mathfrak{d}$。关于此类相似性度量，我们建立了几种数学性质。最后，我们将注意力集中在对特定数字图对的分析上$\rho$ 和 $\nu$ 文献中对它们进行了经典的研究，并且对于每种方法，我们都展示了与前述制粒细化有关的主要性能。