Bipolar semicopulas

The concept of semicopula plays a fundamental role in the aggregation theory on interval [0,1][0,1]. Semicopulas are applied, for example, in the definition of universal integrals. We present an extension of the notion of semicopula to the case of symmetric bipolar interval [−1,1][−1,1]. We call this extension bipolar semicopula. The last definition can be used to obtain a simplified definition of the bipolar universal integral. Moreover, bipolar semicopulas allow for an extension of theory of quasi-copulas to the interval [−1,1][−1,1].