有限Kleene代数自由生成的MV-代数
MV-algebras freely generated by finite Kleene algebras
摘要：
如果V和W是代数簇，使得任意V-代数A在W中有一个约化U（A），则存在一个作用于A的遗忘函子U：V->W|-> U（A）在对象上，并且在同态上相同。根据一般的考虑，这个函子U总是有一个左伴随F：W->V.我们称F（B）为W-代数B自由生成的V-代数。在这个广泛的背景下，自然会出现两个问题。描述问题是用W-代数B的结构尽可能明确地描述V-代数F（B）的结构。识别问题是在一个给定的V-代数A的结构上找到一个W-代数B存在的充分必要条件，使得F（B）与A同构。在有限分配格自由生成的MV-代数的基础上，本文给出了当V是MV-代数的簇，W是Kleene代数的簇，B是等价有限的时，MV-代数的描述和识别问题的解决方案.这些证明在很大程度上依赖于Kleene代数的Davey-Werner自然对偶，MV-代数的紧致有理多面体表示，以及MV-代数的基理论。
英文摘要：
If V and W are varieties of algebras such that any V-algebra A has a reduct U(A) in W, there is a forgetful functor U: V->W that acts by A |-> U(A) on objects, and identically on homomorphisms. This functor U always has a left adjoint F: W->V by general considerations. One calls F(B) the V-algebra freely generated by the W-algebra B. Two problems arise naturally in this broad setting. The description problem is to describe the structure of the V-algebra F(B) as explicitly as possible in terms of the structure of the W-algebra B. The recognition problem is to find conditions on the structure of a given V-algebra A that are necessary and sufficient for the existence of a W-algebra B such that F(B) is isomorphic to A. Building on and extending previous work on MV-algebras freely generated by finite distributive lattices, in this paper we provide solutions to the description and recognition problems in case V is the variety of MV-algebras, W is the variety of Kleene algebras, and B is finitely generated--equivalently, finite. The proofs rely heavily on the Davey-Werner natural duality for Kleene algebras, on the representation of finitely presented MV-algebras by compact rational polyhedra, and on the theory of bases of MV-algebras.
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