This paper introduces and discusses a new algebraic structure, the quasi-topologic structure. The idea of this structure comes from language analysis on the one hand and from analysis of some real situations of clustering on the other. From the cognitive point of view, it is related to the Logic of Determination of Objects (LDO) and to the Logic of Typical and Atypical Objects (LTA) which is particular case of LDO. From the mathematical point of view, it is related to topology. By introducing the notion of internal and external border, it extends the notion of border from classical topology.
The difference between typical instances and atypical instances in a natural categorization process has been introduced by E. Rosh and studied by cognitive psychology and AI. A lot of the knowledge representation systems are expressed in using fuzzy concepts but a degree of membership raises some problem for natural categorizations (especially to classification problems in anthropology, ethnology, archeology, linguistics but also in ontologies), but atypical instances of a concept cannot be apprehended adequately by different degrees from a prototype.
This paper provides a framework to construct a computational model of conceptual metaphor. We first analyze how conceptual metaphor is described by Algebraic Semiotic at linguistic level and by Institutional Theory (an abstract model theory) at a general logical level. By the Logic of Determination of Objects, which has been used in a system of semantic annotation and in a building ontologies system, we further provide a new computational model as a rival approach.