On systems of equations with unknown multifunctions related to the plurality function

Abstract

Let $T$ be a nonempty set. Inspired by a problem posed by Z. Moszner in [10] we investigate for which additional assumptions put on multifunctions $Z(t):T → 2^{ℝ(m)}$, which fulfil condition \[ ⋃_{t ∈ T} Z(t)=ℝ(m)\], and the system of conditions \[Z(t1)^{k1} ∩ Z(t2)^{k2} + Z(t1)^{l1} ∩ Z(t2)^{l2} ⊂ Z(t1)^{k1l1} ∩ Z(t2)^{k2l2}\] for all $t1,t2 ∈ T$ and for all $k1,k2,l1,l2 ∈ \{0,1\}$ such that $k1l1 + k2l2 ≠ 0$, where $ℝ(m) := [0,+∞)^m ∖ \{0_m\}, Z(t)^1 := Z(t), Z(t)^0 := ℝ(m) ∖ Z(t)$, the multifunctions are also satisfying system of equations obtained by replacing the inclusion in the above conditions by the equality. Next we study if this system of equations are equivalent to some system of conditional equations.