Abstract
Aggregation functions on \([0,1]\) with annihilator 0 can be seen as a generalized product on \([0,1]\). We study the generalized product on the bipolar scale \([-1,1]\), stressing the axiomatic point of view ( compare also Greco et al. in IPMU 2014, CCIS 444, Springer International Publishing, New York, 2014). Based on newly introduced bipolar properties, such as the bipolar monotonicity, bipolar unit element, bipolar idempotent element, several kinds of generalized bipolar product are introduced and studied. A special stress is put on bipolar semicopulas, bipolar quasi-copulas and bipolar copulas. Inspired by the truncated sum on \([-1,1]\) we introduce also the class of generalized bipolar sums, which differ from uninorms due to the non-associativity.
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