On Lower Semi-Continuity of Interval-Valued Multihomomorphisms

It is well known that if f is a continuous homomorphism on (R,+), then there exists a constant c ϵ R such that f(ϗ)=cϗ for all ϗ ϵ R. Termwuttipong et al. extended this result to interval-valued multifunctions on R. They proved that an interval-valued multifunction f on R is an upper semi-continuous multihomomorphism on (R,+) if and only if f has one of the following forms : f(ϗ)={cϗ},f(ϗ)=R,f(ϗ)=(0,∞),f(ϗ)=(−∞,0),f(ϗ)=[cϗ,∞),f(ϗ)=(−∞,cϗ] where c is a constant in R. In this paper, we extend the above well known result by considering lower semi-continuity. It is shown that an interval-valued multifunction f on R is a lower semi-continuous multihomomorphism on (R,+) if and only if f is one of the following: f(ϗ)={cϗ},f(ϗ)=R,f(ϗ)=(cϗ,∞),f(ϗ)=(−∞,cx),f(ϗ)=[cϗ,∞),f(ϗ)=(−∞,cϗ] where c is a constant in R.