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Lattice Theory
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
For finite posets the covering relations can be used to obtain graphical representations of the sets and their order relations as was done in Fig. 3.2. In this representation, known as a Hasse diagram, each element of a poset is represented as a vertex in the plane and a line segment or curve that goes upward from x to y whenever y covers x. A segment may cross another segment but must not touch any vertices except at the segment's endpoints. In this text the vertices will be represented by small circles.
Reusing process fragments for fast service composition: a clustering-based approach
Published in Enterprise Information Systems, 2019
From the above formal context, a set of clusters called ‘formal concepts’ are derived to form a complete lattice structure. A lattice of formal concepts can be represented in the form of an ordered diagram, also called a Hasse diagram. Each node of this diagram corresponds to a cluster and the arcs represent the relations of inclusion between the concepts. Each formal concept has two sections. The Extent section regroups a sub-set of objects sharing the common attributes, whereas the Intent section contains those attributes shared by objects in the Extent part. Figure 1 shows an example of concept lattice corresponding to the formal context in Table 2.
Soft cooperation systems and games
Published in International Journal of General Systems, 2018
J. R. Fernández, I. Gallego, A. Jiménez-Losada, M. Ordóñez
We say y covers x if and . The Hasse diagram of the poset is the graph whose vertices are the elements in K, whose links are the cover relations (we have a link between two elements if one of them covers the other one) and if then y is drawn above x. A (maximal saturated) chain in the poset is a sequence of elements in K such that is a bottom, is a top and covers for all . Namely, a chain is a line in the Hasse diagram from a bottom to a top. A poset is named graded if all the chains have the same number of elements. The power set of K is the family of subsets of K and it is denoted by . The poset using the inclusion of sets as order is a boolean algebra.
Cyber-Physical Systems, a new formal paradigm to model redundancy and resiliency
Published in Enterprise Information Systems, 2020
From the above formal context, a set of clusters called ‘formal concepts’ are derived to form a complete lattice structure. A lattice of formal concepts can be represented in the form of an ordered diagram, also called a Hasse diagram. Each node of this diagram corresponds to a cluster and the arcs represent the relations of inclusion between the concepts. Each formal concept has two sections. The Extent section regroups a sub-set of objects sharing the common attributes, whereas the Intent section contains those attributes shared by objects in the Extent part. The figure 3 shows an example of concept lattice corresponding to the formal context in Table 1