Algebraic structures on commutative coquantales

In this paper, we present the notion of commutative coquantale frames as logical relational systems. We introduce an approach to study these frames through the utilization of distance functions in place of the commonly used fuzzy partially orders. We show that a commutative coquantale frame can be regarded as an algebraic structure that induces a commutative coquantale, while a commutative coquantale with q-distance function leads to the development of a commutative coquantale frame. Moreover, we provide several examples pertaining to these notions.