On the Semigroup of Semi-Continuous Interval-Valued Multihomomorphisms

A characterization of semi-continuous interval-valued multihomomorphisms on (R,+) has been given as follows: An interval-valued multifunction f on R is a semi-continuous multihomomorphism on (R,+) if and only if f is one of the following forms: f(x)={cx}, f(x)=R, f(x)=(0,∞), f(x)=(−∞,0), f(x)=[cx,∞) and f(x)=(−∞,cx] where c is a constant in R. Denote by SIM(R,+) the set of all such multifunctions on R. We show that SIM(R,+) is a semigroup under composition and it is a regular semigroup.