JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (6): 76-83.doi: 10.6040/j.issn.1671-9352.0.2022.491

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Sub-Z-Quantales and their properties

Ling WANG(),Bin ZHAO*()   

  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710119, Shaanxi, China
  • Received:2022-09-16 Online:2024-06-20 Published:2024-06-17
  • Contact: Bin ZHAO E-mail:wangling123202208@163.com;zhaobin@snnu.edu.cn

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Abstract:

Firstly, the concept of the sub-Z-Quantales is introduced and some properties of sub-Z-Quantales are studied. In particular, the binary operation ⊙ on the set of all sub-Z-Quantales of a unital Z-Quantale containing the identity element is constructed such that the set is a Quantale. Secondly, the definition of join sub-Z-Quantales is given, and it is proved that every join sub-Z-Quantale of a Z-Quantale with a maximum (minimum) element has a maximum (minimum) element. Finally, the concept of conuclei on Z-Quantales is introduced, and it is proved that the join sub-Z-Quantales and conuclei on a Z-Quantale are one-to-one correspondence.

Key words: Z-Quantale, sub-Z-Quantale, conucleus

CLC Number: 

  • O153.1

Fig.1

The binary relation on Q1"

Table 1

The binary operation * on Q1"

* 0 a b c
0 0 0 0 0
a 0 a b c
b 0 b b c
c 0 c c c

Fig.2

The binary relation on Q3"

Table 2

The binary operation * on Q3"

* a b c d
a a a a a
b a b b b
c a b c c
d a b c d

Fig.3

The binary relation on Q4"

Table 3

The binary operation * on Q4"

* a b c d e
a a a a a a
b a b c d d
c a c c d d
d a d d d d
e a d d d d

Fig.4

The binary relation on Q5"

Table 4

The binary operation * on Q5"

* a b c
a a b b
b b b b
c b b c

Fig.5

The binary relation on Q6"

Fig.6

The binary relation on Q7"

Table 5

The binary operation * on Q7"

* a b c d
a a a a a
b a b c d
c a c c d
d a d d d
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[1] LIU Min, ZHAO Bin*. Localic nuclei and conuclei on Quantales [J]. J4, 2013, 48(2): 81-87.
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