L-subquantales and L-filters of quantales based on quasi-coincidence of fuzzy points with parameters

Abstract

We have investigated the fuzziness of quantales by means of
quasi-coincidence of  fuzzy points with two parameters based on
$L$-sets and have developed two more generalized fuzzy structures, called
$(\in_g, \in_g\vee q_h)$-$L$-subquantales and $(\in_g, \in_g\vee
q_h)$-$L$-filters, respectively. Concretely, some
intrinsic connections between $(\in_g, \in_g\vee
q_h)$-$L$-subquantales and crisp subquantales have first been established. Furthermore, employing the
new characterizations of $(\in_g, \in_g\vee q_h)$-$L$-filters of
quantales, we have studied the relationships between $(\in_g, \in_g\vee
q_h)$-$L$-filters of quantales and their extensions and especially the
essential connections between $(\in_g, \in_g\vee
q_h)$-$L$-subquantales and $(\in_g, \in_g\vee q_h)$-$L$-filters of
quantales. At the same time, the sufficient conditions
of the extension of an $(\in_g, \in_g\vee
q_h)$-$L$-filter being an $(\in_g, \in_g\vee q_h)$-$L$-filter of a
quantale have also been offered. At last, with applying category theory to $(\in_g, \in_g\vee
q_h)$ $L$-subquantales (resp., $L$-filters) of quantales, we have
discovered that the category {\bf GLFquant} (resp., {\bf GFFQant}) of
$(\in_g, \in_g\vee q_h)$ $L$-subquantales (resp., $L$-filters) is of
a topological construct on {\bf Quant} and posses equalizers and
pullbacks.