Concept lattices and similarity in non-commutative fuzzy logic
Concept lattices and similarity in non-commutative fuzzy logic
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发表时间：
2002-05
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通讯作者：
G. Georgescu;A. Popescu
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作者：
G. Georgescu;A. Popescu
经典(清晰)概念由其范围(一组对象)和其意图(一组属性)给出。在交换模糊逻辑中，考虑到对象和属性的模糊集，泛化是自然而然的。在这两种情况下(第一种情况实际上是第二种情况的一种特殊情况)，情况是完全对称的：概念由一对(A，B)给出，其中A是共享来自B的属性的最大对象集合，而B是来自A的对象共享的最大属性集合(涉及到模糊性时具有必要的细微差别)。由于这种对称性，使用对象与使用属性相同，因此不需要做出任何选择。在本文中，我们将概念定义在“非交换模糊世界”中，在这个世界中，句子的连接不一定是可交换的，这导致了以下非对称情况：一个概念具有一个范围(因为，归根结底，概念是用某些描述来包含不同的对象集合)，而是由非交换连接的两个残余物(含义)给出的两个意图。
A classical (crisp) concept is given by its extent (a set of objects) and its intent (a set of properties). In commutative fuzzy logic, the generalization comes naturally, considering fuzzy sets of objects and properties. In both cases (the first being actually a particular case of the second), the situation is perfectly symmetrical: a concept is given by a pair (A,B), where A is the largest set of objects sharing the attributes from B and B is the largest set of attributes shared by the objects from A (with the necessary nuance when fuzziness is concerned). Because of this symmetry, working with objects is the same as working with properties, so there is no need to make any choice. In this paper, we define concepts in a "non-commutative fuzzy world", where conjunction of sentences is not necessarily commutative, which leads to the following non-symmetrical situation: a concept has one extent (because, at the end of the day, concepts are meant to embrace, using certain descriptions, diverse sets of objects), but two intents, given by the two residua (implications) of the non-commutative conjunction.