Bipolar semicopulas

The concept of semicopula plays a fundamental role in the definition of a universal integral. We present an extension of semicopula to the case of symmetric interval [−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 extension of theory of copulas to the interval [−1,1]. 1 Bipolar semicopulas Definition 1. A semicopula is a function⊗ : [0,1]× [0,1]→ [0,1], which is nondecreasing and has 1 as neutral element, i.e. – if a1 ≤ a2 and b1 ≤ b2, then a1⊗ b1 ≤ a2⊗ b2; and – 1⊗ a = a⊗ 1= a. Note that a semicopula has 0 as annihilator. Indeed 0 ≤ a⊗ 0 ≤ 1⊗ 0 = 0 and 0 ≤ 0⊗ a≤ 0⊗ 1= 0. Definition 2. A bipolar semicopula is a function ⊗b : [−1,1] → [−1,1] that is “absolute-nondecreasing”, has 1 as neutral element and−1 as opposite-neutral element, and preserves the sign rule, i.e (A1) if |a1| ≤ |a2| and |b1| ≤ |b2| then |a1⊗b b1| ≤ |a2⊗b b2|; (A2) a⊗b±1=±1⊗b a =±a; and (A3) sign(a⊗b b) = sign(a)⊗b sign(b). Let us note that a bipolar semicopula also satisfies the following additional properties (A4) a⊗b 0= 0⊗b a = 0; (A5) sign(a)⊗b sign(b) = sign(a ·b); and (A6) |a⊗b b|= |a|⊗b |b|.