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In this paper, we introduce the notion of bipolar-valued neutrosophic subbisemiring (BVNSBS), level sets of BVNSBS, and bipolar valued neutrosophic normal subbisemiring (BVNNSBS) of a bisemiring. The concept of BVNSBS is a new generalization of subbisemiring over bisemirings. We discussed the theory of (ξ, τ )- BVNSBS and (ξ, τ )-BVNNSBS over bisemirings and presented several illustrative examples to demonstrate the sufficiency and validity of the proposed theorems, lemmas, and propositions

Pythagorean neutrosophic soft set (PNSS set) is a new approach towards decision making under uncertainty. The PNSS set has much stronger abilities than the neutrosophic soft set and the Pythagorean fuzzy soft set. In this paper, we discuss aggregated operations for aggregating the PNSS decision matrix. The TOPSIS and VIKOR methods are strong approaches for multi criteria group decision making (MCGDM), which is various extensions of neutrosophic soft sets. In this approach, we propose a score function based on aggregating TOPSIS and VIKOR methods to the PNSS-positive ideal solution and the PNSS-negative ideal solution. Also, the TOPSIS and VIKOR methods provide the weights of decision-making. Afterward, a revised closeness is introduced to identify the optimal alternative.&nbsp

We introduce the notion of neutrosophic subbisemiring(shortly NSBS), level sets of NSBS and neutrosophic normal subbisemiring(NNSBS) of a bisemiring. The concept of neutrosophic subbisemiring is a new generalization of fuzzy subbisemiring over bisemiring. We interact the theory for (α, β) NSBS and NNSBS over bisemiring. Let A be the neutrosophic subset in S, we show that ˜̟ = (̟T A, ̟I A, ̟F A) is an NSBS of S if and only if all non empty level set ˜̟(t,s) is a subbisemiring of S for t, s ∈ [0, 1]. Let A be the NSBS of a bisemiring S and V be the strongest neutrosophic relation of S, we observe that A is an NSBS of S if and only if V is an NSBS of S × S. Let A1, A2, ..., An be the family of NSBSs of S1, S2, ..., Sn respectively. We show that A1 × A2 × ... × An is an NSBS of S1 × S2 × ... × Sn. The homomorphic image of NSBS is an NSBS. The homomorphic preimage of NSBS is an NSBS. Examples are provided to illustrate our results

We introduce the notion of neutrosophic subbisemiring(shortly NSBS), level sets of NSBS and neutrosophic normal subbisemiring(NNSBS) of a bisemiring. The concept of neutrosophic subbisemiring is a new generalization of fuzzy subbisemiring over bisemiring. We interact the theory for (��, ��) NSBS and NNSBS over bisemiring. Let A be the neutrosophic subset in S, we show that ���� = (��T A, ��I A, ��F A) is an NSBS of S if and only if all non empty level set ����(t,s) is a subbisemiring of S for t, s ��� [0, 1]. Let A be the NSBS of a bisemiring S and V be the strongest neutrosophic relation of S, we observe that A is an NSBS of S if and only if V is an NSBS of S �� S. Let A1, A2, ..., An be the family of NSBSs of S1, S2, ..., Sn respectively. We show that A1 �� A2 �� ... �� An is an NSBS of S1 �� S2 �� ... �� Sn. The homomorphic image of NSBS is an NSBS. The homomorphic preimage of NSBS is an NSBS. Examples are provided to illustrate our results