On AFS algebra––Part I
Introduction
Since L.A. Zadeh introduced the fuzzy set and theory in 1965, it has received much attention and found successful applications in various fields. In utilizing fuzzy set theory, one should first determine the membership functions of fuzzy sets, since they are the expressions of fuzzy sets in essence. So far, however, determining the membership function of fuzzy sets is usually influenced by personal subjectively factors in some concrete situations, it lacks in the uniformity and completeness from the view of mathematics. Therefore, how to obtain and determine the membership functions properly is an important research topic.
Motivated by the aforementioned concerns, AFS structure and AFS algebras were first proposed by Liu [1], a new axiomatic system based on AFS structure and AFS algebras is established. In this approach, membership functions are described by strict and uniform mathematical methods, and each fuzzy concept can be represented by an element of the AFS algebras in some AFS structure. Recently, the topological molecular lattice structures and topological structures are also established on AFS structures by Liu [2], [3], thus the relations between concepts and some other variable properties of concepts can be described by modern mathematical theory, such as topological molecular lattices and topology, and the abstractions of human cognitive activities can be obtained through the AFS theory.
In this paper, we continue study the topological structure of the AFS structures on the basis of papers [1], [2], [3]. Explore the properties and structures of EI algebra and EIn algebra, and formulate and prove results about them.
The paper is organized as follows. Section 2 reviews both the definitions and results in papers [1], [2], [3], [4], [5], [6], [7]. In Section 3, some new properties of lattice (EM,⩾) are discussed, establish the structure of set of irreducible elements in EM, formulate the standard minimal family of an element in EM, and prove that (EM,⩾) is a new type of molecular lattice, which is neither Boolean algebra nor fuzzy lattice. In Section 4, explore properties of lattice (EX1…XnM,⩾) just the same as those of EI algebra. Finally conclusions are given in Section 5.
Section snippets
Preliminaries
Definition 2.1 Let L be a complete lattice, a∈L, and B⊂L. B is called a minimal family of a if B≠∅ and satisfies the following conditions: supB=a. ∀A⊂L, supA⩾a implies that ∀x∈B, there exists y∈A such that y⩾x.[4]
Definition 2.2 [4]
Let L be a complete lattice, a∈L, B⊂L. B is said to be a standard minimal family of a if B is a minimal family of a and members of B are ∨-irreducible elements.
Theorem 2.1 [4]
Let L be a complete lattice. Then L is a molecular lattice (i.e. completely distributive lattice) if and only if ∀a∈L, a has a standard minimal
On EI algebra
In this section, we will further discuss the properties of lattice (EM,⩾) on the basis of the concept and structure of EI algebra introduced by Liu [1], and mainly deal with the following three problems:
- (1)
The structure of set of irreducible elements in EM will be given.
- (2)
The standard minimal family of an element in EM will be formulated.
- (3)
To prove that (EM,⩾) is a new type of molecular lattice, which is neither a Boolean algebra nor a fuzzy lattice.
Definition 3.1 [1]
Let M be a non-vacuous set, the EI algebra on M,
On EIn algebra
It is known that EIn algebra is a general algebra, which includes EI algebra. Of course, the algebraic structure of EIn algebra is more complicated than that of EI algebra. In this section, similar to study of EI algebra, we will make a thorough study of the properties of lattice (EX1⋯XnM,⩾).
The following proposition is straightforward by Definition 2.13. Proposition 4.1 Let X1,…,Xn,M be n+1 non-vacuous sets, α=∑i∈I(u1i⋯uniAi), β=∑j∈J(v1j⋯vnjBj), and α,β∈EX1⋯XnM, we have the following properties: , α+β=β+α
Conclusions
In this paper, we have discussed some properties and structures of EI algebra and EIn algebra, and obtained the main results as follows:
- •
The expressions of *-irreducible elements (or ∧-irreducible elements) in molecular lattice (EM,⩾) and (EX1⋯XnM,⩾) are proposed, respectively.
- •
The standard minimal family of elements in molecular lattice (EM,⩾) and (EX1⋯XnM,⩾) have been formulated, respectively.
- •
That neither molecular lattice (EM,⩾) nor molecular lattice (EX1⋯XnM,⩾) is a fuzzy lattice is proved.
Acknowledgements
This research has been supported by National Key Basic Research Development Programming of China (No. 2002CB312200) and Science Fund of China (No. 60174014).
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