AF inverse monoids and the structure of countable MV-algebras

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• 作者：Mark V. Lawson^a ; ^{m.v.lawson@hw.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Philip Scott^b ; ^{phil@site.uottawa.ca" class="auth_mail" title="E-mail the corresponding author}
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：221
• 期：1
• 页码：45-74
• 全文大小：670 K

文摘

This paper is a further contribution to the developing theory of Boolean inverse monoids. These monoids should be regarded as non-commutative generalizations of Boolean algebras; indeed, classical Stone duality can be generalized to this non-commutative setting to yield a duality between Boolean inverse monoids and a class of étale topological groupoids. MV-algebras are also generalizations of Boolean algebras which arise from many-valued logics. It is the goal of this paper to show how these two generalizations are connected. To do this, we define a special class of Boolean inverse monoids having the property that their lattices of principal ideals naturally form an MV-algebra. We say that an arbitrary MV-algebra can be co-ordinatized if it is isomorphic to an MV-algebra arising in this way. Our main theorem is that every countable MV-algebra can be so co-ordinatized. The particular Boolean inverse monoids needed to establish this result are examples of what we term AF inverse monoids and are the inverse monoid analogues of AF C^⁎$C^{⁎}$-algebras. In particular, they are constructed from Bratteli diagrams as direct limits of finite direct products of finite symmetric inverse monoids.