Introduction to Neutrosophic Measure, Neutrosophic Integral, and Neutrosophic Probability

Florentin Smarandache

Source-linked AI summary

Introduction to Neutrosophic Measure, Neutrosophic Integral, and Neutrosophic Probability

Florentin Smarandache

arXiv:1311.7139v1Nov 27, 2013cs.AI

TL;DR

The paper addresses the absence of formal neutrosophic measures and integrals while extending neutrosophic probability to represent indeterminacy alongside randomness. It defines these concepts and shows that neutrosophic probability generalizes classical probability, coinciding with it when indeterminacy is zero.

Problem
The paper addresses the lack of a scientific notion of neutrosophic measure and related formal tools for representing indeterminacy.
Method
It defines neutrosophic measures and integrals, develops neutrosophic probability, and illustrates the framework with practical examples.
Results
Neutrosophic probability analyzes random phenomena together with related indeterminacy and coincides with classical probability when indeterminacy is zero.
Takeaways & Limitations
The framework supports multiple definitions of neutrosophic measures, integrals, and probabilities according to the type of indeterminacy involved.
Takeaways & Limitations
Treating indeterminacy as zero by convention is possible in some games but is not fully accurate.

Abstract

from arXiv · show

In this paper, we introduce for the first time the notions of neutrosophic measure and neutrosophic integral, and we develop the 1995 notion of neutrosophic probability. We present many practical examples. It is possible to define the neutrosophic measure and consequently the neutrosophic integral and neutrosophic probability in many ways, because there are various types of indeterminacies, depending on the problem we need to solve. Neutrosophics study the indeterminacy. Indeterminacy is different from randomness. It can be caused by physical space materials and type of construction, by items involved in the space, etc.

The author · Chapter 1. Introduction to Neutrosophic Measure

The paper introduces neutrosophic measure as a framework that represents determinate, indeterminate, and opposite components, then defines its spaces, functions, probability measures, properties, generalizations, and examples. It emphasizes that indeterminacy can be modeled separately from truth, falsehood, or ordinary measurement outcomes.

1.1. Introduction.: Neutrosophic concepts distinguish an item A, its opposite antiA, and a neutral or indeterminate component neutA.Examples include victory/defeat/tie, truth/falsehood/indeterminacy, and voting for, against, or not voting for a candidate.
1.2. Definition of Neutrosophic Measure.: The paper introduces neutrosophic measure as a function ν(A) = (m(A), m(neutA), m(antiA)) on a neutrosophic space and σ-neutrosophic algebra.The three components measure the determinate part of A, its indeterminate part, and the determinate part of its opposite.
1.2. Definition of Neutrosophic Measure.: The measure assigns the empty set (0,0,0) and satisfies countable additivity for countable collections of disjoint neutrosophic sets.The definition also treats absent opposite or neutral components as having null measure.
1.3. Neutrosophic Measure Space: A neutrosophic measure space is the triplet (X,Σ,ν), with normalized measures satisfying ν(X) = (x_1,x_2,x_3), x_1 + x_2 + x_3 = 1, and x_1,x_2,x_3 ≥ 0.Measurable neutrosophic sets are members of Σ, and their inverse images under neutrosophic measurable functions remain measurable.
1.5. Finite Neutrosophic Measure Space: The framework defines finite and σ-finite neutrosophic measures, non-negativity, and continuity from below or above under the stated sequence and finiteness conditions.Without non-negativity, a measure satisfying only null-empty-set and countable-additivity axioms takes at most one of the ±∞ values.
1.10. Neutrosophic Probability Measure: Neutrosophic probability measures probable or possible propositions, use the range 0^- ≤ ν(X) ≤ 3^+, and distinguish absolute from relative measures.They are denoted NP to connect them with classical probability P.
1.11. Neutrosophic Category Theory: The theory forms a neutrosophic category whose objects are neutrosophic spaces and whose arrows are structure-preserving measurable mappings satisfying associativity and identity-unit axioms.The paper also states monotonicity and additivity properties for neutrosophic measures.

Chapter 2: Introduction to Neutrosophic Integral

This chapter defines the neutrosophic integral with respect to a neutrosophic measure and explains how indeterminacy in functions, limits, spaces, and measures makes the integral approximate.

Definition: The neutrosophic integral is defined using a neutrosophic measure on a neutrosophic measure space X.The integral is taken with respect to the neutrosophic measure ν.
Sources of indeterminacy: Integration-related indeterminacy may concern the function value, integration limits, or the space and its measure.
Neutrosophic functions: A neutrosophic function combines a determinate part g(x) with an indeterminate part i(x), making its values approximate.The indeterminate component is constrained by i(x) ∈ [0, h(x)] with h(x) ≥ 0, and fN(x) is represented by an interval involving g(x) and h(x).
Integral approximation: The neutrosophic integral is likewise approximate and is expressed as the sum of the determinate integral and the indeterminate integral.
Indeterminate integration limits: When the lower limit a is uncertain, it can be modeled with a determinate part a1 and an indeterminate part ε, yielding alternative indeterminacy intervals.The example assigns indeterminacies i1 and i2 to intervals associated with the uncertain lower limit.

Chapter 3: Introduction to Neutrosophic Probability … 3.16. Neutrosophic Random Variables.

The paper extends probability to account for indeterminacy arising from imperfect physical spaces, objects, information, and event interpretations. Neutrosophic probability therefore models randomness and indeterminacy through neutrosophic random variables that may be discrete, continuous, or mixed.

3.1. First Example of Indeterminacy.: Cracked or irregular surfaces can create an indeterminate die outcome in addition to faces 1–6, making the neutrosophic sample space include indeterminacy.For a die tossed on a cracked surface, the die may rest on an edge or vertex, and the neutrosophic probability of a face is less than 1/6 because there are seven possible outcomes.
3.2. Second Example of Indeterminacy; 3.3. Third Example of Indeterminacy; 3.4. Fourth Example of Indeterminacy; 3.5. Fifth Example of Indeterminacy; 3.6. Sixth Example of Indeterminacy; 3.7. The Seventh Example of Indeterminacy; 3.8. The Eighth Example of Indeterminacy; 3.9. Ninth Example of Indeterminacy.: Erased die faces, coins stuck in cracks, damaged ballots, blank or rejecting votes, ambiguous weather, unreliable tests, unreadable roulette numbers, and damaged cards all generate indeterminacy.Examples distinguish indeterminate outcomes from ordinary complementary probabilities and show that physical defects or incomplete information can prevent exact outcome identification.
3.10. Tenth Example of Indeterminacy.: Neutrosophic probability represents soccer outcomes as winning, tying, or losing, rather than only winning versus not winning.The chapter also notes a rare indeterminate sex outcome among newborns when sex is ambiguous.
3.12. Example of a Neutrosophic Continuous Random Variable.: A spinner illustrates a neutrosophic continuous random variable whose outcome may fall in an interval or an indeterminate zone.The supplied passage identifies an indeterminate-zone probability of 1/4.
3.15. Distinction between Indeterminacy and Randomness.: Neutrosophic probability differs from randomness because it analyzes random phenomena together with indeterminacy caused by defective spaces or imperfect physical objects.It consequently uses random variables and indeterminacy variables, along with stochastic and indeterminate processes.
3.16. Neutrosophic Random Variables.: A neutrosophic random variable changes through both randomness and indeterminacy, and its values represent possible outcomes and possible indeterminacies.These variables can involve objective or subjective randomness and indeterminacy and can be classified as discrete, continuous, or mixed.
3.16. Neutrosophic Random Variables.: Neutrosophic random variables may be finite or infinite, with infinite cases being countable or uncountable, and admissibility requires computing chance, indeterminacy, and nonchance over ranges.The range must map to a subset of the neutrosophic sample space.

3.17. Many Possible Neutrosophic Measures and Probabilities. … 3.25. Neutrosophic Probability Axioms.

The paper defines neutrosophic probability as a context-dependent, three-component generalization of classical and imprecise probability that represents occurrence, indeterminacy, and non-occurrence. It extends probability spaces, mass functions, and axioms to neutrosophic sample spaces and events.

3.17. Many Possible Neutrosophic Measures and Probabilities.: Neutrosophic measures and probabilities may be defined in multiple ways because approximations and indeterminacies depend on each application.
3.18. Definition of Neutrosophic Probability: Neutrosophic probability estimates event occurrence, related indeterminacy, and non-occurrence, while neutrosophic variables and processes can have unclear outcomes over time.It generalizes classical probability and coincides with it when stochastic-process indeterminacy is zero.
3.19. Neutrosophic Probability vs. Imprecise Probability.: Neutrosophic probability generalizes imprecise probability by representing an event with truth, indeterminacy, and falsehood components rather than only an interval.
3.20. Sigma-Algebra of Events.: A sigma-algebra is closed under the empty set, whole space, complements, and countable unions, and classical probability is additive over disjoint events.
3.22. Neutrosophic Sigma-Algebra of Events.: The neutrosophic sigma-algebra follows the same structure while allowing the sample space to contain indeterminate parts.
3.23. Neutrosophic Probability Measure.: The neutrosophic probability measure maps a neutrosophic sample space to three-component probabilities and permits their total to be below, equal to, or above 1.The components correspond to occurrence, indeterminacy, and anti-occurrence, with the whole-space sum constrained between -0 and 3+.
3.24. Neutrosophic Probability Mass Function.: A neutrosophic event is a subset of the neutrosophic sample space, and its probability mass function sums occurrence, indeterminacy, and anti-occurrence contributions.
3.25. Neutrosophic Probability Axioms.: Neutrosophic probability axioms extend Kolmogorov axioms to a space comprising a neutrosophic sample space, event space, and probability measure, including σ-additivity for disjoint events.The axioms also support neutrosophic addition, inclusion-exclusion, and set-difference formulas; relaxing the third axiom yields a quasiprobability distribution.

3.27. Interpretations of the Neutrosophic Probability. … 3.32. Neutrosophic Example with Tossing a Coin Multiple Times.

These sections define neutrosophic probability spaces and events, distinguish objective and subjective interpretations, and illustrate frequentist calculations under indeterminacy. Examples with dice and repeated coin flips show product-space construction, multiple indeterminacy types, propagation, and reduced classical-event probability.

3.27. Interpretations of the Neutrosophic Probability.: Neutrosophic probability has objective and subjective interpretations, respectively describing states of affairs and degrees of belief in events.
3.28. Neutrosophic Notions.: A neutrosophic experiment produces indeterminacy, whose complete outcomes form a neutrosophic sample space and whose collections define neutrosophic events.
3.29. Example with Neutrosophic Frequentist Probability.: Frequentist neutrosophic probability uses simulation and chance components for an event, its opposite, and indeterminacy, with corresponding formulas for unions and intersections.
3.30. Example with Neutrosophic Frequentist Probability on a Neutrosophic Product Space.: The same product-space example reports neutrosophic probability values 0.0225, 0.1900, and 0.7875 for the indicated event and indeterminacy components.
3.31. Example with Double Indeterminacy.: A die with erased faces introduces physical-die and physical-space indeterminacies, with total indeterminacy 0.40 and an equivalent four-faced die on an irregular surface.
3.32. Neutrosophic Example with Tossing a Coin Multiple Times.: For a fair coin with edge-sticking chance 0.02, three flips generate first-, second-, and third-order indeterminacy, totaling 27 elements and demonstrating that indeterminacy propagates.
3.32. Neutrosophic Example with Tossing a Coin Multiple Times.: The HTT outcome has neutrosophic probability (0.117649, 0.058808, 0.823543), and 0.117649 < 0.125000 because indeterminacy has strictly positive chance.

3.33. Example with Sum of Chances of an Event. … 3.36. Neutrosophic Mutually Exclusive Events.

These sections extend neutrosophic probability from summing outcome chances to handling contradictory, incomplete, and mutually exclusive or overlapping events. Examples show how indeterminacy is represented alongside chance and how event unions account for overlap and unknown information.

3.33. Example with Sum of Chances of an Event.: For A = {a₁, a₂, …, aₙ}, neutrosophic probability records the sum of outcome chances together with indeterminacy and complement-chance components.The event probability is represented as a neutrosophic triplet rather than a single scalar.
3.33. Example with Sum of Chances of an Event.: In the die example, each outcome has neutrosophic probability (0.15, 0.10, 0.75) when the chance of indeterminacy is 0.10.The same triplet is assigned to outcomes 1, 2, and 3 in the example.
3.34. Paraconsistent Neutrosophic Probability.: Paraconsistent neutrosophic probability has components whose sum is strictly greater than 1, producing contradictions between chances.Different forecasting criteria can yield different chances of occurrence for the same event.
3.34. Paraconsistent Neutrosophic Probability.: The handball example assigns G a 60% chance from historical disputes, H a 70% chance from current-season performance, and a 10% tie chance from a compensatory criterion.The values arise from different criteria applied to the same forthcoming game.
3.35. Incomplete Neutrosophic Probability.: Incomplete neutrosophic probability has components whose sum is strictly less than 1, indicating incomplete or missing information.In the revised handball example, each weak-performing team has a 20% winning chance and a tie has a 30% chance.
3.36. Neutrosophic Mutually Exclusive Events.: For mutually exclusive neutrosophic events, the framework gives an analogous union property while retaining an indeterminacy component.The section also states a corresponding formulation for non-mutually-exclusive neutrosophic events.

3.37. Neutrosophic Experimental Probability. … 3.45. De Morgan’s Neutrosophic Laws.

The paper extends probability concepts to neutrosophic settings by incorporating indeterminacy into experimental, conditional, Bayesian, multiplicative, and complementary-event formulations.

3.37. Neutrosophic Experimental Probability.: Neutrosophic experimental probability is introduced alongside classical experimental probability, with indeterminate outcomes represented separately from event occurrences and non-occurrences.The supplied formulas count occurrences, indeterminate occurrences, and non-occurrences over the total number of trials.
3.38. Neutrosophic Survey.: A neutrosophic survey provides a way to obtain neutrosophic experimental probability.An example tosses a regular die five times on an irregular surface and records 2, 5, 1, indeterminacy, and 4.
3.39. Neutrosophic Conditional Probability for Independent Events.: For neutrosophically independent events, conditional chance equals the corresponding unconditional chance for an event, its indeterminacy, and its complement.The supplied equations state ch(A given B) = ch(A), ch(indeterminacy given B) = ch(indeterminacy), and ch(complement of A given B) = ch(complement of A).
3.42. Neutrosophic Bayesian Rule.: The neutrosophic Bayesian Rule is presented as a generalization involving event chance, indeterminacy, and complement terms conditioned on B.The supplied expression explicitly includes ch(B), ch(indeterminacy A | B), and ch(complement A | B).
3.43. Neutrosophic Multiplicative Rule.: The neutrosophic Multiplication Rule combines event chances with conditional chances and adds indeterminacy terms for conjunctions and alternatives.Its displayed formula includes ch(A) · ch(B given A), indeterminacy conjunction and disjunction terms, and complement-related conditional terms.
3.43. Neutrosophic Multiplicative Rule.: The indeterminacy component in the multiplicative rule is reduced to an inclusion-exclusion-style expression involving unconditional and conditional indeterminacy.The supplied derivation gives ch(indeterminacy) + ch(indeterminacy | A) − ch(indeterminacy) · ch(indeterminacy | A).
3.44. Neutrosophic Negation (or Neutrosophic Probability of Complementary Events).: For an event A different from indeterminacy, neutrosophic negation defines the complement probability using the sample-space chance, complement indeterminacy, and the original event chance.The supplied double-negation expression is ch(anti(antiA)) = (ch(X)-ch(A), ch(indetermantiA), ch(A)).

3.46. Neutrosophic Double Negation. … 3.50. Neutrosophic Discrete Probability Spaces.

These sections extend neutrosophic probability through double negation, expected values, product spaces, and applications involving soccer outcomes and indeterminate dice. They also show how conventions and generalized discrete products accommodate indeterminacy.

3.46. Neutrosophic Double Negation.: 3.46. For events different from indeterminacy, neutrosophic double negation preserves the event probability: NP(anti(antiA)) = NP(A).The result is expressed componentwise as (ch(A), ch(indetermA), ch(antiA)).
3.47. Neutrosophic Expected Value.: 3.47. Neutrosophic expected value combines numerical outcomes associated with determined chances and indeterminacy chances.The example assigns outcomes of losing $2.00 for an A-vote, gaining $3.00 for a B-vote, and losing $1.00 for an indeterminate vote.
3.48. Neutrosophic Probability and Neutrosophic Logic Used in The Soccer Games.: 3.48. Neutrosophic probability represents soccer games through chances to win, tie, and lose, unlike classical or imprecise probabilities described here as representing one result only.For the stated example, Alpha has 0.6 chance to win, 0.2 chance to tie, and 0.1 chance to lose.
3.48. Neutrosophic Probability and Neutrosophic Logic Used in The Soccer Games.: 3.48. The soccer example can also use neutrosophic logical “and” and fuzzy “or” operators, including min/max or product and probabilistic-sum forms.The paper notes that neutrosophic logic’s three components may sum to a value different from 1.
3.49. A Neutrosophic Question.: 3.49. The two-dice sum-of-6 problem exposes ambiguity over whether adding indeterminacy should preserve 6 or produce indeterminacy.The paper states that treating indeterminacy as zero is a possible convention but not quite true.
3.49. A Neutrosophic Question.: 3.49. With a single-die indeterminacy chance of 0.10 and chance of each determined face 0.15, the neutrosophic probability of sum = 6 is (0.1125, 0.1900, 0.6975).The components correspond to chance of sum = 6, chance of indeterminate sum = 6, and chance of sum ≠ 6 with no indeterminacy.
3.50. Neutrosophic Discrete Probability Spaces.: 3.50. Neutrosophic discrete probability products combine event chances across two spaces and generalize to s spaces using subset-based inclusion-exclusion terms.The construction indexes subsets of {1, 2, … , s} whose cardinality is t, for 1 ≤ t ≤ s.

3.51. Classification of Neutrosophic Probabilities. … 3.54. Numerical Example of Fusion of Subjective Neutrosophic Probabilities:

The sections distinguish objective, frequentist, and subjective neutrosophic probabilities, then present a counting principle and a source-fusion procedure. A numerical example applies the fusion formulas and concludes that the satellite is more likely friendly.

3.51. Classification of Neutrosophic Probabilities.: Objective neutrosophic probability applies when the chances of all events, including indeterminacy, can be computed objectively.The unreadable-face die illustrates an objectively computable setup.
3.51. Classification of Neutrosophic Probabilities.: Frequentist neutrosophic probability uses experiments when some event or indeterminacy chance cannot be computed exactly.Repeated die tossing estimates indeterminacy as favorable cases over total cases, but results are approximate and can differ across repetitions.
3.51. Classification of Neutrosophic Probabilities.: Subjective neutrosophic probability applies when neither exact computation nor frequentist experimentation is possible.Different sources may assign different estimates, such as 60% friend, 30% hostile, and 10% neutral.
3.52. The Fundamental Neutrosophic Counting Principle.: For sequential neutrosophic events, E followed by F has e·f ways, e1·f+e·f1 first-order indeterminacies, and e1·f1 second-order indeterminacies.The section also gives a die-based sample-space illustration of the principle.
3.53. A Formula for the Fusion of Subjective Neutrosophic Probabilities.: Combining more sources of information is proposed to improve subjective neutrosophic probability approximations.The fusion example considers satellite states that are friendly (t), neutral (i), or hostile (f).
3.53. A Formula for the Fusion of Subjective Neutrosophic Probabilities.: The fusion procedure redistributes cross-products among truth, indeterminacy, and falsehood proportionally to the corresponding source probabilities.The passages describe redistributions for t1·i2, t2·i1, truth–falsehood, and indeterminacy–falsehood combinations, using normalized or non-normalized inputs.
3.54. Numerical Example of Fusion of Subjective Neutrosophic Probabilities:: The numerical example substitutes the observers’ values into the fusion formulas and obtains a final expression of ≃0.40903.The example then states that the satellite has a higher chance of being friendly.

3.55. General Formula for Fusioning Classical Subjective Probabilities Provided by Two Sources. … 3.59. Removing Indeterminacy.

The section develops fusion procedures for subjective probabilities from multiple sources, including PCR5 redistribution and neutrosophic logic inference. It also distinguishes aggregation from logical conjunction and examines removing or refining indeterminacy in neutrosophic probability spaces.

3.55. General Formula for Fusioning Classical Subjective Probabilities Provided by Two Sources.: PCR5 redistributes conflicting chances back to the corresponding alternatives when combining subjective probabilities from two information sources.The redistribution principle is identified with the PCR5 rule used in information fusion.
3.56. Different Ways of Combining Neutrosophic Subjective Probabilities Provided by Two Sources.: Normalized neutrosophic probabilities for friendly, neutral, and hostile outcomes satisfy F1 + N1 + H1 = F2 + N2 + H2 = 1.The combined probabilities likewise satisfy F + N + H = 1.
3.56. Different Ways of Combining Neutrosophic Subjective Probabilities Provided by Two Sources.: The nine pairwise terms from two sources are distributed among F, N, and H, with conflicting friendly-hostile terms assigned either to N or proportionally back to F and H.The two choices represent pessimistic or less pessimistic treatments, while alternative assignments represent less optimistic or very optimistic treatments.
3.57. Neutrosophic Logic Inference type in Fusioning Subjective Neutrosophic Probabilities.: Neutrosophic logic inference applies when an aircraft must be classified as friendly, neutral, or hostile using subjective chances from two sources.The probability space is defined as Φ = {F, N, H}, with empty intersections of events.
F N H: The F N H inference combines pessimistic and optimistic results using t-norm and t-conorm operations, with normalization when needed.The text suggests comparing or averaging results obtained from other AND/OR operators.
3.58. Neutrosophic Logic vs. Subjective Neutrosophic Probability.: Subjective neutrosophic probability aggregates chances from multiple information sources within one probability space, whereas neutrosophic logic computes conjunctions of propositions.The distinction concerns aggregation procedures for source-provided chances versus logical conjunction.
3.59. Removing Indeterminacy.: Removing indeterminacy produces an incomplete classical sample space in which the first and third Kolmogorov axioms remain valid but the second fails.The neutrosophic probability of the whole sample space becomes strictly less than 1.
3.59. Removing Indeterminacy.: n-Valued refined neutrosophic probability represents truth, indeterminacy, and falsity through indexed property-specific components, but refinements depend on the event and chosen properties.The text states that some events cannot have their occurrence, non-occurrence, or related indeterminacy refined, while refinements may be performed in multiple ways.

3.61. Neutrosophic Markov Chain. · P R D · 3.62. Applications of Neutrosophics.

The paper extends classical Markov chains by incorporating indeterminacy into state transitions and illustrates the framework with a three-state world-economy example. It also defines neutrosophic probability operations and identifies broad scientific and humanistic applications involving indeterminacy, unknowns, or neutrality.

3.61. Neutrosophic Markov Chain.: Neutrosophic Markov chains generalize classical Markov chains by incorporating indeterminacy into the probability space.They are characterized as memoryless systems in which the next state depends only on the current state.
3.61. Neutrosophic Markov Chain.: Higher-order neutrosophic Markov chains allow the next state to depend on the previous m states rather than only the current state.The paper also distinguishes discrete-time definitions from continuous-time formulations using a continuous index.
3.61. Neutrosophic Markov Chain.: The world-economy example uses prosperity (P), recession (R), and depression (D) as the Markov-chain states.Its transition graph assigns separate frequencies to known, unknown, and negative transition outcomes.
P R D: The state space in the P R D example is {P, R, D}, with stochastic row vectors representing the individual states.The construction normalizes the rows of the resulting matrix.
P R D: Neutrosophic probabilities are combined through defined multiplication and addition operators, with alternative operators also possible.The paper specifies component-wise operations and notes that operator choices can vary.
P R D: When neutrosophic probability reduces to classical probability, the neutrosophic probability matrix gives the same result as the classical probability matrix.The two-year calculation includes the result (0.1742, 0.05, 0.02).
P R D: After two years, the largest chance for the economy is being in recession.This conclusion follows from the normalized neutrosophic transition matrix.
3.62. Applications of Neutrosophics.: Neutrosophics can be applied in statistical physics, financial markets, risk management, mathematical biology, quantum theory, and other fields involving indeterminacy, unknowns, or neutrality.The paper denotes neutrality with respect to an item A as <neutA>.

Chapter 4. Neutrosophic Subjects for Future Research

The chapter outlines future research directions spanning neutrosophic mathematical structures, relational and graph-based models, information fusion, and applications. It specifically identifies topics including neutrosophic numbers, rough sets, matrices, graphs, trees, and relational databases.

Mathematical structures: Future work includes neutrosophic topologies, metric spaces, smooth topological spaces, numbers, arithmetic operations, rough sets, and relational structures.Neutrosophic numbers use a+bI with I representing indeterminacy and I^2 = I; proposed work also includes ranking procedures.
Relational and graph models: Additional directions include neutrosophic relational maps, matrices ranging from bimatrices to n-matrices, graphs, and trees with at least one indeterminate edge or node.The graph and tree definitions explicitly require an indeterminate edge or indeterminate node.
Applications: The chapter also proposes neutrosophic fusion rules for information fusion and applications in relational databases and image processing.Listed image-processing applications include thresholding, denoising, and segmentation.

ADDENDA

The addenda lists additional publications extending neutrosophic logic, sets, topology, and methods, alongside applications in image processing, robotics, semantic services, and decision making. It also records work on paradoxes, research methodology, and domain-specific problem solving.

Logic and Set Theory: Additional work develops neutrosophic logic and set theory through N-norms, N-conorms, topologies, and n-ary operators.The listed works address operators and topological structures in neutrosophic logic and sets.
Applications: Applications include image segmentation, robotics, semantic web services, information fusion, and qualitative causal reasoning.Examples span ultrasound and watershed segmentation, robotics, semantic-web agents, information-fusion proceedings, and complex-system reasoning.
Decision Making: Several publications apply neutrosophic models to multicriteria decision making using aggregation operators, cross-entropy, correlation coefficients, and similarity measures.The bibliography lists methods for simplified, single-valued, and interval neutrosophic sets.
Conceptual and Methodological Extensions: The addenda also includes studies of paradoxicity, neutrosophic diagrams, S-denying theories, time-travel paradoxes, and neutrosophic research methods.These entries address conceptual and methodological extensions within neutrosophics.