A Study On The Modality, Assignment And Complementarity

Lattice theory, as a branch of algebra, is defined as a logical operation on the basis of a set system. In order to make the traditional logic system more accurate and easy to operate, George Boole introduced a series of mathematical symbols and operation rules to the traditional logic system, which leads to the generation of lattice theory in the 1840s. Now lattice theory has got a wide range of application in many subjects such as Data Mining, Distributed Computation, Electronic Engineering and Computer Science.This paper summarizes the distributive, modular and complementary properties of lattices and classifies lattices from perspectives of three concepts—modular lattices, distributive lattices and complemented lattices. By observing classification, some questions as follows are put forward. In what condition can non-uniquely complemented lattices be distinguished by whether they are modular or non-modular lattices? Are there uniquely comparable complemented non-modular lattices? In the process of classification, a question that whether there exists a uniquely complemented non-modular lattice will be encountered. By referring to previous works, it has been found that this question has engendered broad thoughts of many mathematicians, and the exploration of and discussion on this issue have promoted the advancement and development of lattice theory. This paper makes a simple combination of the development of the question, conducts further research on the existence theorem of the uniquely complemented non-modular lattices (i.e. Dilworth’s Theorem), and through simplifying three-element-lattice direct product lattices, in the original process of proof, to two-element-lattice direct product lattices, the theorem can also be proved. Some properties of the direct product of lattices are studied. Explore and study modular properties of complemented lattices and at the same time, use the property of direct product, construct the uniquely comparable complemented lattices and uniquely comparable complemented non-modular lattices respectively. This shows that uniquely comparable complemented lattices can be distinguished, according to whether it is a modular lattice, into uniquely comparable complemented modular lattices and uniquely comparable complemented non-modular lattices. Only by additional condition that "there excludes comparable complement in the lattice" cannot modular and non-modular lattices be distinguished. Through the discussion of the more general situation, this paper draws some conclusion on how to distinguish non-uniquely complemented lattices into non-uniquely complemented modular lattices and non-uniquely complemented non-modular lattices.