J4 ›› 2013, Vol. 48 ›› Issue (2): 81-87.

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Localic nuclei and conuclei on Quantales

LIU Min, ZHAO Bin*   

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062,  Shaanxi, China
  • Received:2012-03-01 Online:2013-02-20 Published:2013-03-04

Abstract:

This paper deals with localic nuclei and conuclei on quantales and the unital quantale Q[e]. Firstly, it is proved that a localic nucleus on a quantale Q can be uniquely extended to the unital quantale Q[e]. Also, it is proved that a twosided commutative quantale has a largest localic subquantale, which is applied to study the extension of the largest localic quantic conucleus on Q to the unitale quantale Q[e]. All the extensions of a largest localic quantic conucleus to the unitale quantale Q[e] are given. And it is proved that Q[e] has a largest localic subquantale if and only if Q is the trivial quantale.

Key words: Quantale; Locale; unital quantale; Quantic nucleus; Quantic conucleus

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