序半群Quantale完备化的进一步结果

本文给出了序半群上幂集Quantale的最大拓扑闭包的具体形式,即序半群的最小Quantale完备化.在此基础上,证明了Quantale范畴Quant是序半群范畴OSGrp*的满的反射子范畴.最后,给出了序半群Quantale完备化的一个应用.     

序半群的定向完备化

首先引入了序半群定向完备化的概念;其次证明了序半群的序理想之集和理想之集都是连续定向完备序半群,进而序半群的序理想之集和负序半群的理想之集是它们的定向完备化;最后得到了序半群范畴的几个满子范畴的反射子范畴。     

Quantale中的素理想及弱素理想

本文把Quantale中的序结构与代数运算＆结合在一起给出了Quantale中素理想和弱素理想的概念。讨论了它们之间的关系，得到了Quantale中理想是(弱)素理想的充要条件。证明了与序半群中的一些经典结论相一致的命题。     

Quantale中理想的扩张

讨论理想Quantale的性质,给出了当Q是可换Quantale时,Q中理想都是半素理想的一个条件.引入了理想的扩张的概念,证明了与序半群中的一些经典结论相一致的命题.通过理想的扩张构造了一个Quantale上的同余,得到了当原理想是素理想时,这个同余所确定的Quantale商是Frame且找到了它的具体结构.     

Quantale中的素根定理

在Quantale中引入了m系的概念,利用m系讨论了Quantale中素理想和半素理想之间的关系.在此基础上证明了当Q是可换Quantale时,Id(Q)是空间式Quantale当且仅当Q中的任一理想都是半素理想.最后把环和序半群中的素根定理推广到Quantak中,得到了Quantale中的素根定理.     

广义算子半群范数连续问题研究

研究了Hilbert空间上范数连续广义算子半群的特征条件.利用广义半群的的预解式,给出了广义算子半群范数连续的充分条件.     

L-Fuzzy闭包算子的扩张原理

Ｆｕｚｚｙ闭包算子的扩张原理揭示了Ｆｕｚｚｙ闭包算子与经典闭包算子之间的密切关系,是利用传统学科已有结论研究Ｆｕｚｚｙ数学相关理论的有效工具。本文讨论了当Ｌ为有限分配格时,Ｌ－Ｆｕｚｚｙ闭包算子与闭包系统的扩张问题,并给出一种具体的由经典闭包算子生成Ｌ－Ｆｕｚｚｙ闭包算子的方法及其部分性质。     

广义算子半群的扰动定理

算子扰动问题是研究微分方程的一个重要工具,首先结合Hilbert空间中有界算子引导的广义算子半群的定义研究了广义半群的性质;其次重点讨论了广义算子半群的扰动问题,给出了广义算子半群的加法扰动定理成立的条件.     

L-偏序集中L_k-素闭包算子和L_k-素内部算子

本文在完备剩余格上引入了L-偏序集,给出了L-偏序集上L_k-素闭包算子和L_k-素内部算子的概念及其等价刻画,在此基础上推广得出了n重L_k-素闭包算子和n重L_k-素内部算子的概念及其等价刻画。     

Hilbert空间中广义算子半群范数连续的特征

研究了Hilbert空间上最终范数连续广义算子半群的特征条件,利用半群的生成元的预解式,给出了Hilbert空间上广义算子半群范数连续的三个特征条件.     

Precoherent quantale completions of partially ordered semigroups

Just as complete lattices can be viewed as the completions of posets, quantales can also be treated as the completions of partially ordered semigroups. Motivated by the study on the well-known Frink completions of posets, it is natural to consider the “Frink” completions for the case of partially ordered semigroups. For this purpose, we firstly introduce the notion of precoherent quantale completions of partially ordered semigroups, and construct the concrete forms of three types of precoherent quantale completions of a partially ordered semigroup. Moreover, we obtain a sufficient and necessary condition of the Frink completion on a partially ordered semigroup being a precoherent quantale completion. Finally, we investigate the injectivity in the category $$\mathbf {APoSgr}_{\le }$$ of algebraic partially ordered semigroups and their submultiplicative directed-supremum-preserving maps, and show that the $$\mathscr {E}_{\le }$$-injective objects of algebraic partially ordered semigroups are precisely the precoherent quantales, here $$\mathscr {E}_{\le }$$ denote the class of morphisms $$h:A\longrightarrow B$$ that preserve the compact elements and satisfy that $$h(a_1)\cdots h(a_n)\le h(b)$$ always implies $$a_1\cdots a_n\le b$$.     

Quantum B-algebras

The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor “spectrum”, it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale Q is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deficiency of the inverse residuals of Q. The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete.     

Generation results forB-bounded semigroups

In [3]A. Bellini-Morante defined and analysed a new one-parameter family of bounded operators which he called a B-bounded semigroup. The definition was motivated by an example from the transport theory where the evolution generated by an operator A was in a certain sense controlled by another operator B. In this paper we show that a given pair (A, B) generates a B-bounded semigroup if and only if in a certain extrapolation space related to the operator B, the closure of A generates a semigroup and we also address some related topics.     

Ordinary semicascades and their ergodic properties

A relationship is considered between ergodic properties of a discrete dynamical system on a compact metric space Ω and characteristics of companion algebro-topological objects, namely, the Ellis enveloping semigroup E, the Köhler enveloping operator semigroup Γ, and the semigroup G being the closure of the convex hull of Γ in the weak-star topology on the operator space EndC*(Ω). The main results are formulated for ordinary (having metrizable semigroup E) semicascades and for tame dynamical systems determined by the condition cardE ? c. A classification of compact semicascades in terms of topological properties of the semigroups specified above is given.     

QUantale中的闭滤子

在Quantale中讨论了与Gabriel拓扑密切关联的闭滤子,给出了闭滤子与闭映射之间的相互确定关系．证明了凝聚左侧Quantale Q的闭滤子全体F(Q)在包含序下构成Frame,并且Q的紧闭滤子全体Fc(Q)是F(Q)的子Frame．最后,证明了凝聚双侧交换Quantale的紧闭滤子全体在包含序下构成Coherent Frame．     

Value semigroups,value quantales,and positivity domains

In 1981 and 1997 Kopperman and Flagg, respectively, proved that every topological space is metrisable, provided the symmetry and separation axioms are removed from the requirements on the metric, and the metric is allowed to take values in, respectively, a value semigroup or a value quantale. Seeking to construct a value quantale from a value semigroup we focus on a small portion of the structure present in a value semigroup, comprising what we call a positivity domain, and we construct its enveloping value quantale, forming part of a detailed comparison between value semigroups and value quantales. We obtain a representation theorem for value quantales in terms of positivity domains, and we outline how products of positivity domains can be used in the theory of continuity spaces instead of (the non-existent) products of value quantales.     

Completions for partially ordered semigroups

A standard completion γ assigns a closure system to each partially ordered set in such a way that the point closures are precisely the (order-theoretical) principal ideals. If S is a partially ordered semigroup such that all left and all right translations are γ-continuous (i.e., Y∈γS implies {x∈S:y·x∈Y}∈γS and {x∈S:x·y∈Y}∈γS for all y∈S), then S is called a γ-semigroup. If S is a γ-semigroup, then the completion γS is a complete residuated semigroup, and the canonical principal ideal embedding of S in γS is a semigroup homomorphism. We investigate the universal properties of γ-semigroup completions and find that under rather weak conditions on γ, the category of complete residuated semigroups is a reflective subcategory of the category of γ-semigroups. Our results apply, for example, to the Dedekind-MacNeille completion by cuts, but also to certain join-completions associated with so-called “subset systems”. Related facts are derived for conditional completions. A first draft of this paper by the second author, containing parts of Section 2, was received on August 9, 1985.     

Generalized Kantorovich Operators on Bauer Simplices and Their Limit Semigroups

In this paper, we prove an asymptotic formula for generalized Kantorovich operators associated with the canonical Markov projection on a given Bauer simplex K. That formula involves an operator acting on the subalgebra of all products of a?ne functions on K. Moreover, we prove that such an operator is closable and its closure is the generator of a Markov semigroup which, in turn, may be represented in terms of iterates of the above-mentioned generalized Kantorovich operators.     

On the category Q-Mod

In this paper we consider the category Q-Mod of modules over a given quantale Q. The paper is motivated by constructions and results from the category of modules over a ring. We show that the category Q-Mod is monadic, consider its relation to the category Q-Top of Q-topological spaces and generalize a method of completion of partially ordered sets. Received December 20, 2005; accepted in final form December 4, 2006.