On bipolar-valued subbisemirings of bisemirings and their extension

We defined bipolar-valued subbisemirings, level sets of bipolar-valued subbisemirings, and bipolar-valued normal subbisemirings of bisemirings. Additionally, we look into some of these subbisemirings related properties (shortly, SBS). Let A be a bipolar-valued fuzzy set (BVFS) in S. Prove that (f) over tilde = < f(A)(p); f(A)(n)> is a bipolar-valued subbisemiring of S if and only if all non-empty level set (f) over tilde ((t,s)) is a subbisemiring of S for t is an element of [0, 1] and s is an element of [- 1, 0]. Let A be a BVSBS of a bisemiring S and V be the strongest bipolar-valued relation of S. Prove that A is a BVSBS of S if and only if V is a BVSBS of S x S. The homomorphic image and pre-image of BVSBS are also BVSBS. Let f (alpha) over tilde be an (alpha, beta)-BVSBS of S. Prove that the nonempty sets fpff and fn ff are SBSs of S, where f(alpha)(p) = {p is an element of S |f(p)(p) > alpha(p)} and fn ff = {p is an element of S | f(n)(p) < alpha(n)}. Let <(f)over tilde> =< f(A)(p); f(A)(n)> be any BVFS in S. Prove that (f) over tilde is an (alpha, beta)-BVSBS of S if and only if each non-empty level subset (f) over tilde ((t,s)) is an SBS of S for all t is an element of (alpha(p), beta(p)] and s. (alpha(n), beta(n)]. Examples are given to demonstrate our findings.