Through this article, the definitions of an L -fuzzy pseudo-metric and a family of pseudo-metric that satisfies the property of lower-semicontinuity (or named nests of pseudo-metrics) are introduced. After that, we indicated that an L -fuzzy pseudo-metric can be constructed from nests of pseudo-metrics and nests of pseudo-metrics can be constructed from an L -fuzzy pseudo-metric. Therefore, we establish a each to each association between L - fuzzy pseudo-metrics and nests of pseudo-metrics.

In this paper, we firstly analyse the position relation on four points from four cases, that is, (i) both x and y are on the outside side of m and n, (ii) either x or y is on the inner side of m and n, (iii) both x and y are on the inner side of m and n, (iv) both x and y are on the same side of m and n. Then we acquire eight kinds of transitivity-like postulates from the four cases. What is more, we discuss the interrelationships between eight transitivity-like postulates and get some important deductibility theorems. Finally, we give a few counter-examples to show that the inverse of the deductibility theorems do not hold and get a summary diagram to show the relationships among these transitivity-like postulates.