Fuzzy prime filters of lattice implication algebras

In this paper, some properties of fuzzy filters are given. Besides, the structure of fuzzy filters are further studied. And finally, the concept of fuzzy prime filter is proposed with some equivalent conditions of fuzzy prime filters obtained.     

Permuting operations on strings and their relation to prime numbers

Some length-preserving operations on strings only permute the symbol positions in strings; such an operation X gives rise to a family {X_n}_n≥2 of similar permutations. We investigate the structure and the order of the cyclic group generated by X_n. We call an integer n X-prime if X_n consists of a single cycle of length n (n≥2). Then we show some properties of these X-primes, particularly, how X-primes are related to X^′-primes as well as to ordinary prime numbers. Here X and X^′ range over well-known examples (reversal, cyclic shift, shuffle, twist) and some new ones based on the Archimedes spiral and on the Josephus problem.     

Fuzzy Boolean algebras

The purpose of this paper is to generalize the following situation: from the concrete structure $\text{B}$, we define the notion of Boolean algebras; the Stone representation theorem allows us to replace the algebraic study of Boolean algebras by a topological one. Let E be a non-empty set, and J a non-empty ordered set. Note $\text{B}$ the set of all fuzzy subsets of (E,J). We shall introduce the concept of fuzzy Boolean algebra and find a representation theorem. But it will be difficult to speak of the dual fuzzy topological space of a fuzzy Boolean algebra as we shall see further, except in certain particular cases.     

Coloring infinite graphs and the Boolean prime ideal theorem

It is shown that the following theorem holds in set theory without AC: There is a functionG which assigns to each Boolean algebraB a graphG(B) such that (1) ifG(B) is 3-colorable then there is a prime ideal inB and (2) every finite subgraph ofG(B) is 3-colorable. The proof uses a combinatorial lemma on finite graphs.     

模糊软布尔代数

将模糊软集与布尔代数相结合,定义了模糊软布尔代数、模糊软布尔子代数、模糊理想软布尔代数、模糊软布尔代数的模糊软同态的概念,并研究了它们的性质.     

布尔代数的Fuzzy子代数和Fuzzy理想

引入了布尔代数的Fuzzy子代数、Fuzzy理想和Fuzzy商布尔代数的概念,给出了布尔代数的Fuzzy集是Fuzzy子代数(Fuzzy理想)的充要条件,讨论了布尔代数的Fuzzy子代数(Fuzzy理想)在布尔代数同态下的像和逆像,得到了布尔代数的Fuzzy子代数的同态基本定理。     

The Boolean prime ideal theorem and products of cofinite topologies

We show:

1. The Boolean Prime Ideal theorem is equivalent to each one of the statements:
   1. “For every family of compact spaces, for every family of basic closed sets of the product with the fip there is a family of subbasic closed sets () with the fip such that for every ”.
   2. “For every compact Loeb space (the family of all non empty closed subsets of has a choice function) and for every set X the product is compact”.
2. (: the axiom of choice restricted to families of finite sets) implies “every well ordered product of cofinite topologies is compact” and “every well ordered basic open cover of a product of cofinite topologies has a finite subcover”.
3. (: the axiom of choice restricted to countable families of finite sets) iff “every countable product of cofinite topologies is compact”.
4. (: every filter of extends to an ultrafilter) is equivalent to the proposition “for every compact Loeb space having a base of size and for every set X of size the product is compact”.

     

布尔代数的双极值模糊子代数

在布尔代数中引入了双极值模糊子代数的概念,讨论了布尔代数的双极值模糊子代数的性质.证明了布尔代数的两个双极值模糊子代数交与直积也是双极值模糊子代数.     

Completel-groupoids and their prime spectra

We argue for the validity of the infinite ∩-distributivity law in a lattice of radical elements of an arbitrary complete l-groupoid, in particular, in the lattice of radical ideals of an arbitrary ring. The notion of a prime spectrumSpec (L) is defined, and we point out a specific place that topological spaces of typeSpec (L) hold among all T_o-spaces. Supported by the AMS grant. Translated fromAlgebra i logika, Vol. 36, No. 3, pp. 341–355, May–June, 1997.     

Fuzzy filters of BE-algebras

Characterizations of fuzzy filters in a BE-algebra are established. Conditions for a fuzzy set to be a fuzzy filter are given. For a fuzzy set µ the least fuzzy filter containing µ is constructed. The homomorphic properties of fuzzy filters of a BE-algebra are provided. Finally, characterizations of Noetherian BE-algebras and Artinian BE-algebras via fuzzy filters are obtained.     

布尔代数的直觉T-S模糊子代数及理想

在布尔代数中引入了的直觉T-S模糊子代数和直觉T-S模糊理想的概念,给出了布尔代数的直觉T-S模糊子代数的两个等价定义,进一步讨论了它们的性质.证明了布尔代数的两个直觉T-S模糊子代数(理想)的模交与直积也是直觉T-S模糊子代数(理想).     

布尔代数上的Fuzzy同余关系

引入了布尔代数上的Fuzzy同余关系的概念,讨论了布尔代数上的Fuzzy同余关系与布尔代数的Fuzzy理想之间的关系,给出了商布尔代数的同构定理。     

数据库中布尔型及广义模糊加权关联规则的挖掘

引入布尔型加权关联规则和广义模糊加权关联规则的概念,并分别给出挖掘这些规则的计算方法．     

Thin-tall Boolean algebras and their automorphism groups

Our main result is a general construction to show that for each countable groupG there are 2¹ pairwise non-isomorphic thin-tall superatomic Boolean algebras, all of them representingG as a group of non-trivial automorphisms.Presented by J. Mycielski.Research supported by the NSERC of Canada.     

Effectively dense Boolean algebras and their applications

A computably enumerable Boolean algebra is effectively dense if for each we can effectively determine an such that implies . We give an interpretation of true arithmetic in the theory of the lattice of computably enumerable ideals of such a Boolean algebra. As an application, we also obtain an interpretation of true arithmetic in all theories of intervals of (the lattice of computably enumerable sets under inclusion) which are not Boolean algebras. We derive a similar result for theories of certain initial intervals of subrecursive degree structures, where is the degree of a set of relatively small complexity, for instance a set in exponential time.

     

Complexity of Boolean algebras and their Scott rank

We deal with problems associated with Scott ranks of Boolean algebras. The Scott rank can be treated as some measure of complexity of an algebraic system. Our aim is to propound and justify the procedure which, given any countable Boolean algebra, will allow us to construct a Boolean algebra of a small Scott rank that has the same natural algebraic complexity as has the initial algebra. In particular, we show that the Scott rank does not always serve as a good measure of complexity for the class of Boolean algebras. We also study into the question as to whether or not a Boolean algebra of a big Scott rank can be decomposed into direct summands with intermediate ranks. Examples are furnished in which Boolean algebras have an arbitrarily big Scott rank such that direct summands in them either have a same rank or a fixed small one, and summands of intermediate ranks are altogether missing. This series of examples indicates, in particular, that there may be no nontrivial mutual evaluations for the Scott and Frechet ranks on a class of countable Boolean algebras. Supported by RFFR grant No. 99-01-00485, by a grant for Young Scientists from SO RAN, 1997, and by the Federal Research Program (FRP) “Integration”. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 643–666, November–December, 1999.