几种格上拓扑空间范畴中乘积与上积运算的封闭性

本文引入四种格上拓扑空间范畴,分别讨论了其中的乘积与上积运算,以及相应的结构性、唯一性和存在性问题。通过讨论,给出了一种比较理想的格上拓扑空间的乘积空间,并指出目前使用的格上拓扑空间的乘积空间具有一定的局限性,其所属范畴的态射是Ｚａｄｅｈ型映射,这类映射保持Ｆｕｚｚｙ点的高度不变,隐含度量不变性。     

Some Properties of Categories CL and CLPair

Structures of products, coproducts, subobjects, extremal subobjects, quotient objects and extremal quotient objects in CL (the category of closed-set lattices) and CLPair (the category of closed-set lattice pairs) are given.      

区间值模糊滤子范畴

讨论区间值模糊滤子范畴IVF和Hausdorff区间值模糊滤子范畴IVFHau中乘积、余积、等子、余等子存在性及构造。证明IVF和IVFHau都不是完备范畴而IVF是余完备范畴。也证明IVF不仅有许多弱拓扑满子范畴,也有许多非弱拓扑满子范畴。     

When ∏ is Isomorphic to ⊕

For a category , we investigate the problem of when the coproduct ⊕ and the product functor ∏ from  ^I to  are isomorphic for a fixed set I, or, equivalently, when the two functors are Frobenius functors. We show that for an Ab category  this happens if and only if the set I is finite (and even in a much general case, if there is a morphism in  that is invertible with respect to addition). However, we show that ⊕ and ∏ are always isomorphic on a suitable subcategory of  ^I which is isomorphic to  ^I but is not a full subcategory. If  is only a preadditive category, then we give an example that shows that the two functors can be isomorphic for infinite sets I. For the module category case, we provide a different proof to display an interesting connection to the notion of Frobenius corings.