能被代表的偏序集

Q1 Mathematics

Journal of Applied Logic Pub Date : 2016-07-01 DOI:10.1016/j.jal.2016.03.003

Rob Egrot

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引用次数: 7

摘要

一个偏序集是可表示的，如果它能以这样一种方式嵌入到集合域中，即现有的有限相遇和连接分别成为交和并(我们说有限相遇和连接是保留的)。更一般地说，对于基数α和β，一个偏序集被称为(α，β)-可表示，如果一个嵌入到集合的域存在，它保留小于α的集合的满足和小于β的集合的连接。利用超积/超根论证证明了当2≤α，β≤ω时，(α，β)可表示的偏集是初等的，但在α或β=ω的情况下不具有有限公理化性。我们还证明了具有保留可数或所有满足和连接的表示的偏序集类是伪初等的。

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Representable posets

A poset is representable if it can be embedded in a field of sets in such a way that existing finite meets and joins become intersections and unions respectively (we say finite meets and joins are preserved). More generally, for cardinals α and β a poset is said to be $(α, β)$ -representable if an embedding into a field of sets exists that preserves meets of sets smaller than α and joins of sets smaller than β. We show using an ultraproduct/ultraroot argument that when $2 \leq α, β \leq ω$ the class of $(α, β)$ -representable posets is elementary, but does not have a finite axiomatization in the case where either α or $β = ω$ . We also show that the classes of posets with representations preserving either countable or all meets and joins are pseudoelementary.