FI代数的滤子

研究FI代数的滤子理论，建立F滤子、P滤子、Q滤子、C滤子之间的联系．     

BL代数的(∈,∈∨q)-模糊滤子理论

对BL代数的(∈,∈∨q)-模糊滤子理论作系统研究。首先,在BL代数中引入(∈,∈∨q)-模糊对合滤子和(∈,∈∨q)-模糊结合滤子两类新概念,获得了这两类(∈,∈∨q)-模糊滤子的几个等价刻画。其次,详细讨论了BL代数中各类(∈,∈∨q)-模糊滤子间的关系,证明了一个模糊集为(∈,∈∨q)-模糊布尔(关联)滤子当且仅当它既是(∈,∈∨q)-模糊正关联滤子又是(∈,∈∨q)-模糊对合滤子。最后,以直观图示的方式对BL代数中各类(∈,∈∨q)-模糊滤子间关系进行了总结。     

关联BCK代数的P滤子、I滤子及BFI滤子

引入关联BCK代数的P滤子、I滤子及BFI滤子的概念,并得到了一些重要的结论.     

Heyting代数的扩张滤子

运用泛代数与逻辑学的方法和原理对Heyting代数中滤子概念作进一步研究.在Heyting代数H中引入了滤子F关于H的子集A的扩张滤子概念并考察其性质.证明了一个滤子F关于H的所有子集的扩张滤子全体之集构成一个完备Heyting代数且构成一个Stone格.     

PFI代数及其P滤子

文「１」引入了ＦＩ代数及其滤子,文「２」讨论了ＦＩ代数与ＭＶ代数的联系,本文在此基础上引入了正关联ＦＩ代数及其正关联滤子,并得到几个重要性质。     

FI-代数的多种Fuzzy滤子

对FI-代数的各种滤子概念进行fuzzy化,引入fuzzy滤子、fuzzy P-滤子、fuzzy Q-滤子和fuzzy C-滤子等概念并给出它们的若干等价刻画。证明了:(1)每个fuzzy Q-滤子都是fuzzy P-滤子;(2)一个fuzzy集成为fuzzy Q-滤子当且仅当它既是fuzzy P-滤子又是fuzzy C-滤子。还给出了多种滤子的扩张定理以及具体的实例和反例。     

FI代数的模糊滤子

在FI代数(Fuzzy蕴涵代数)上引入了模糊滤子的概念并给出了其等价刻画;探讨了由模糊滤子所诱导的同余关系及商代数,证明了由模糊滤子所诱导的同余关系是完备分配格.     

BL-代数的直觉模糊滤子度

提出BL-代数的直觉模糊滤子度的概念,给出其的等价刻画,并用直觉模糊滤子度来讨论BL-代数上的直觉模糊子集是直觉模糊滤子的程度.对于BL-代数的一个直觉模糊集,建立其(强)截集构成的滤子与其直觉模糊滤子度之间的关系.讨论了直觉模糊子集的交、直积的直觉模糊滤子度以及BL-代数的直觉模糊子集的同态像与逆像的直觉模糊滤子度.     

伪补MS-代数的滤子同余关系

在伪补MS-代数上引入余核滤子和完全滤子的概念,研究伪补MS-代数的余核滤子和完全滤子的性质,获得了余核滤子和完全滤子生成的同余关系的表达式,证明了具有余核滤子的最小同余关系有同余一致性.     

On fuzzy fantastic filters of lattice implication algebras

Fuzzification of a fantastic filter in a lattice implication algebra is considered. Relations among a fuzzy filter, a fuzzy fantastic filter, and fuzzy positive implicative filter are stated. Conditions for a fuzzy filter to be a fuzzy fantastic filter are given. Using the notion of level set, a characterization of a fuzzy fantastic filter is considered. Extension property for fuzzy fantastic filters is established. The notion of normal/maximal fuzzy fantastic filters and complete fuzzy fantastic filters is introduced, and some related properties are investigated.     

On decomposition of pseudo BL-algebras

We show that under some conditions, imposed on coatoms and maximal idempotents of a pseudo BL-algebra, we can decompose a pseudo BL-algebra M as an ordinal sum and we show that then M is linearly ordered. We investigate pseudo BL-algebras with a unique coatom a and with a maximal idempotent, and analyze two main situations: either a ⁿ = a ⁿ⁺¹ holds for some n ≥ 1, or a ⁿ > a ⁿ⁺¹ hold for any n ≥ 1. We note that there exist (subdirectly irreducible) algebras with two coatoms that are not linearly ordered, so the restriction to a single coatom is natural.     

BL代数的等价刻画及更多性质

研究了与H(a)jek的模糊命题演算系统BL相对应的BL代数,提出了仅涉及运算*和→的NBL代数概念并探讨了其有关性质,证明了BL代数与N-BL代数是等价的,进而得到了BL代数更多的性质.     

On the structure of generalized BL-algebras

A generalized BL - algebra (or GBL-algebra for short) is a residuated lattice that satisfies the identities . It is shown that all finite GBL-algebras are commutative, hence they can be constructed by iterating ordinal sums and direct products of Wajsberg hoops. We also observe that the idempotents in a GBL-algebra form a subalgebra of elements that commute with all other elements. Subsequently we construct subdirectly irreducible noncommutative integral GBL-algebras that are not ordinal sums of generalized MV-algebras. We also give equational bases for the varieties generated by such algebras. The construction provides a new way of order-embedding the lattice of -group varieties into the lattice of varieties of integral GBLalgebras. The results of this paper also apply to pseudo-BL algebras. This paper is dedicated to Walter Taylor. Received March 7, 2005; accepted in final form July 25, 2005.     

关于BL-代数的模糊滤子与模糊理想

在BL-代数中引入模糊超滤子和模糊固执滤子的概念,证明了如下条件对于BL-代数的非常数模糊滤子f来说是等价的:(1)f是布尔的和素的,(2)f是蕴涵的和素的,(3)f是超的,(4)f是固执的。应用模糊正蕴涵滤子给出G-代数的若干特征性质。提出BL-代数模糊理想的概念,给出一些重要例子,并通过例子说明在BL-代数中模糊理想一般不能由模糊滤子导出。同时,从模糊理想出发构造了商BL-代数,并建立了相应的同态基本定理。最后,研究了BL-代数的几类模糊理想及其相互关系,给出模糊布尔理想、模糊素理想、模糊超理想的特征性质。     

On the predicate logics of continuous t-norm BL-algebras

Given a class C of t-norm BL-algebras, one may wonder which is the complexity of the set Taut(C) of predicate formulas which are valid in any algebra in C. We first characterize the classes C for which Taut(C) is recursively axiomatizable, and we show that this is the case iff C only consists of the Gödel algebra on [0,1]. We then prove that in all cases except from a finite number Taut(C) is not even arithmetical. Finally we consider predicate monadic logics Taut_M(C) of classes C of t-norm BL-algebras, and we prove that (possibly with finitely many exceptions) they are undecidable.Mathematics Subject Classification (2000): Primary: 03B50, Secondary: 03B47Acknowledgement The author is deeply indebted to Petr Hájek, whose results about the complexity problems of predicate fuzzy logics constitute the main motivation for this paper, and whose suggestions and remarks have been always stimulating. He is also indebted to Matthias Baaz, who pointed out to him a method used in [BCF] for the case of monadic Gödel logic which works with some modifications also in the case of monadic BL logic.     

Kites and pseudo BL-algebras

We investigate a construction of a pseudo BL-algebra out of an ?-group called a kite. We show that many well-known examples of algebras related to fuzzy logics can be obtained in that way. We describe subdirectly irreducible kites. As another application, we exhibit a new countably infinite family of varieties of pseudo BL-algebras covering the variety of Boolean algebras.     

BL-代数的模糊素理想

在BL-代数上,将模糊集理论应用到素理想理论中,引入了模糊素理想的概念,研究了模糊素理想的性质,获得了模糊素理想的若干刻画方法和模糊素理想延拓定理等重要结果,证明了由模糊素理想构造的模糊商BL-代数是一个MV-链.     

Compact representations of BL-algebras

In this paper we define sheaf spaces of BL-algebras (or BL-sheaf spaces), we study completely regular and compact BL-sheaf spaces and compact representations of BL-algebras and, finally, we prove that the category of non-trivial BL-algebras is equivalent with the category of compact local BL-sheaf spaces. Mathematics Subject Classification (2000):08A72, 03G25, 54B40, 06F99, 06D05