A philosopher is someone who practices philosophy. The term "philosopher" comes from the Ancient Greek, φιλόσοφος, meaning "lover of wisdom"; the coining of the term has been attributed to the Greek thinker Pythagoras. In the classical sense, a philosopher was someone who lived according to a certain way of life, focusing on resolving existential questions about the human condition, not someone who discourses upon theories or comments upon authors; these particular brands of philosophy are Hellenistic ones and those who most arduously commit themselves to this lifestyle may be considered philosophers. A philosopher is one who challenges what is thought to be common sense, doesn’t know when to stop asking questions, reexamines the old ways of thought. In a modern sense, a philosopher is an intellectual who has contributed in one or more branches of philosophy, such as aesthetics, epistemology, metaphysics, social theory, political philosophy. A philosopher may be one who worked in the humanities or other sciences which have since split from philosophy proper over the centuries, such as the arts, economics, psychology, anthropology and politics.
The separation of philosophy and science from theology began in Greece during the 6th century BC. Thales, an astronomer and mathematician, was considered by Aristotle to be the first philosopher of the Greek tradition. While Pythagoras coined the word, the first known elaboration on the topic was conducted by Plato. In his Symposium, he concludes. Therefore, the philosopher is one. Therefore, the philosopher in antiquity was one who lives in the constant pursuit of wisdom, living in accordance to that wisdom. Disagreements arose as to what living philosophically entailed; these disagreements gave rise to different Hellenistic schools of philosophy. In consequence, the ancient philosopher thought in a tradition; as the ancient world became schism by philosophical debate, the competition lay in living in a manner that would transform his whole way of living in the world. Among the last of these philosophers was Marcus Aurelius, regarded as a philosopher in the modern sense, but refused to call himself by such a title, since he had a duty to live as an emperor.
According to the Classicist Pierre Hadot, the modern conception of a philosopher and philosophy developed predominately through three changes: The first is the natural inclination of the philosophical mind. Philosophy is a tempting discipline which can carry away the individual in analyzing the universe and abstract theory; the second is the historical change through the Medieval era. With the rise of Christianity, the philosophical way of life was adopted by its theology. Thus, philosophy was divided between a way of life and the conceptual, logical and metaphysical materials to justify that way of life. Philosophy was the servant to theology; the third is the sociological need with the development of the university. The modern university requires professionals to teach. Maintaining itself requires teaching future professionals to replace the current faculty. Therefore, the discipline degrades into a technical language reserved for specialists eschewing its original conception as a way of life.
In the fourth century, the word philosopher began to designate a man or woman who led a monastic life. Gregory of Nyssa, for example, describes how his sister Macrina persuaded their mother to forsake "the distractions of material life" for a life of philosophy. During the Middle Ages, persons who engaged with alchemy was called a philosopher – thus, the Philosopher's Stone. Many philosophers still emerged from the Classical tradition, as saw their philosophy as a way of life. Among the most notable are René Descartes, Baruch Spinoza, Nicolas Malebranche, Gottfried Wilhelm Leibniz. With the rise of the university, the modern conception of philosophy became more prominent. Many of the esteemed philosophers of the eighteenth century and onward have attended and developed their works in university. Early examples include: Immanuel Kant, Johann Gottlieb Fichte, Friedrich Wilhelm Joseph Schelling, Georg Wilhelm Friedrich Hegel. After these individuals, the Classical conception had all but died with the exceptions of Arthur Schopenhauer, Søren Kierkegaard, Friedrich Nietzsche.
The last considerable figure in philosophy to not have followed a strict and orthodox academic regime was Ludwig Wittgenstein. In the modern era, those attaining advanced degrees in philosophy choose to stay in careers within the educational system as part of the wider professionalisation process of the discipline in the 20th century. According to a 1993 study by the National Research Council, 77.1% of the 7,900 holders of a PhD in philosophy who responded were employed in educational institutions. Outside academia, philosophers may employ their writing and reasoning skills in other careers, such as medicine, business, free-lance writing and law; some known French social thinkers are Claude Henri Saint-Simon, Auguste Comte, Émile Durkheim. British social thought, with thinkers such as Herbert Spencer, addressed questions and ideas relating to political economy and social evolution; the political ideals of John Ruskin were a precursor of social economy. Important German philosophers and social thinkers included Immanuel Kant, Georg Wilhelm Friedrich Hegel, Karl Marx, Max Weber, Georg Simmel, Martin Heidegger.
Important Chinese philosophers and social thinke
A people is a plurality of persons considered as a whole, as is the case with an ethnic group or nation, but, distinct from a nation, more abstract, more overtly political. Collectively, for example, the contemporary Frisians and Danes are two related Germanic peoples, while various Middle Eastern ethnic groups are linguistically categorized as Semitic peoples. Various states claim to govern, in the name of the people. Both the Roman Republic and the Roman Empire used the Latin term Senatus Populusque Romanus; this term was fixed to Roman legionary standards, after the Roman Emperors achieved a state of total personal autarchy, they continued to wield their power in the name of the Senate and People of Rome. A People's Republic is a Marxist or socialist one-party state that claims to govern on behalf of the people if it in practice turns out to be a dictatorship. Populism is another umbrella term for various political tendencies that claim to represent the people with an implication that they serve the common people instead of the elite.
Chapter One, Article One of the Charter of the United Nations states that peoples have the right to self-determination. In criminal law, in certain jurisdictions, criminal prosecutions are brought in the name of the People. Several U. S. states, including California and New York, use this style. Citations outside the jurisdictions in question substitute the name of the state for the words "the People" in the case captions. Four states — Massachusetts, Virginia and Kentucky — refer to themselves as the Commonwealth in case captions and legal process. Other states, such as Indiana refer to themselves as the State in case captions and legal process. Outside the United States, criminal trials in Ireland and the Philippines are prosecuted in the name of the people of their respective states; the political theory underlying this format is that criminal prosecutions are brought in the name of the sovereign. S. states, the "people" are judged to be the sovereign as in the United Kingdom and other dependencies of the British Crown, criminal prosecutions are brought in the name of the Crown.
"The people" identifies the entire body of the citizens of a jurisdiction invested with political power or gathered for political purposes. Clan Kinship List of contemporary ethnic groups List of indigenous peoples Nationality Tribe
Willard Van Orman Quine
Willard Van Orman Quine was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century." From 1930 until his death 70 years Quine was continually affiliated with Harvard University in one way or another, first as a student as a professor of philosophy and a teacher of logic and set theory, as a professor emeritus who published or revised several books in retirement. He filled the Edgar Pierce Chair of Philosophy at Harvard from 1956 to 1978. A 2009 poll conducted among analytic philosophers named Quine as the fifth most important philosopher of the past two centuries, he won the first Schock Prize in Logic and Philosophy in 1993 for "his systematical and penetrating discussions of how learning of language and communication are based on available evidence and of the consequences of this for theories on knowledge and linguistic meaning." In 1996 he was awarded the Kyoto Prize in Arts and Philosophy for his "outstanding contributions to the progress of philosophy in the 20th century by proposing numerous theories based on keen insights in logic, philosophy of science and philosophy of language."Quine falls squarely into the analytic philosophy tradition while being the main proponent of the view that philosophy is not conceptual analysis but the abstract branch of the empirical sciences.
His major writings include Two Dogmas of Empiricism, which attacked the distinction between analytic and synthetic propositions and advocated a form of semantic holism, Word and Object, which further developed these positions and introduced Quine's famous indeterminacy of translation thesis, advocating a behaviorist theory of meaning. He developed an influential naturalized epistemology that tried to provide "an improved scientific explanation of how we have developed elaborate scientific theories on the basis of meager sensory input." He is important in philosophy of science for his "systematic attempt to understand science from within the resources of science itself" and for his conception of philosophy as continuous with science. This led to his famous quip that "philosophy of science is philosophy enough." In philosophy of mathematics, he and his Harvard colleague Hilary Putnam developed the "Quine–Putnam indispensability thesis," an argument for the reality of mathematical entities. According to his autobiography, The Time of My Life, Quine grew up in Akron, where he lived with his parents and older brother Robert Cloyd.
His father, Cloyd Robert, was a manufacturing entrepreneur and his mother, Harriett E. was a schoolteacher and a housewife. He received his B. A. in mathematics from Oberlin College in 1930, his Ph. D. in philosophy from Harvard University in 1932. His thesis supervisor was Alfred North Whitehead, he was appointed a Harvard Junior Fellow, which excused him from having to teach for four years. During the academic year 1932–33, he travelled in Europe thanks to a Sheldon fellowship, meeting Polish logicians and members of the Vienna Circle, as well as the logical positivist A. J. Ayer, it was Quine who arranged for Tarski to be invited to the September 1939 Unity of Science Congress in Cambridge, for which Tarski sailed on the last ship to leave Danzig before the Third Reich invaded Poland. Tarski survived the war and worked another 44 years in the US. During World War II, Quine lectured on logic in Brazil, in Portuguese, served in the United States Navy in a military intelligence role, deciphering messages from German submarines, reaching the rank of lieutenant commander.
At Harvard, Quine helped supervise the Harvard graduate theses of, among others, David Lewis, Daniel Dennett, Gilbert Harman, Dagfinn Føllesdal, Hao Wang, Hugues LeBlanc, Henry Hiz and George Myro. For the academic year 1964–1965, Quine was a fellow on the faculty in the Center for Advanced Studies at Wesleyan University. In 1980 Quine received an honorary doctorate from the Faculty of Humanities at Uppsala University, Sweden. Quine was an atheist, he had four children by two marriages. Guitarist Robert Quine was his nephew. In the foreword to the new edition of Word and Object, Quine's student Dagfinn Føllesdal noted that Quine began to lose his memory toward the end of his life; the deterioration of his short-term memory was so severe that he struggled to continue following arguments. Quine had considerable difficulty in his project to make the desired revisions to Word and Object. Before passing away, Quine noted to Morton White, "I do not remember what my illness is called, Althusser or Alzheimer, but since I cannot remember it, it must be Alzheimer."
He passed away from the illness on Christmas Day in 2000. Quine was politically conservative, but the bulk of his writing was in technical areas of philosophy removed from direct political issues, he did, write in defense of several conservative positions: for example, in Quiddities: An Intermittently Philosophical Dictionary, he wrote a defense of moral censorship. Quine's Ph. D. thesis and early publications were on formal set theory. Only after World War II did he, by virtue of seminal papers on ontology and language, emerge as a major philosopher. By the 1960s, he had worked out his "naturalized epistemology" whose aim was to answer all substantive questions of knowledge and meaning using the methods and tools of the natural sciences. Quine roundly rejected the notion that there should be a "first philosophy"
Arabic numerals are the ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It is the most common system for the symbolic representation of numbers in the world today; the Hindu-Arabic numeral system was developed by Indian mathematicians around AD 500 using quite different forms of the numerals. From India, the system was adopted by Arabic mathematicians in Baghdad and passed on to the Arabs farther west; the current form of the numerals developed in North Africa. It was in the North African city of Bejaia that the Italian scholar Fibonacci first encountered the numerals; the use of Arabic numerals spread around the world through European trade and colonialism. The term Arabic numerals is ambiguous, it may be intended to mean the numerals used by Arabs, in which case it refers to the Eastern Arabic numerals. Although the phrase "Arabic numeral" is capitalized, it is sometimes written in lower case: for instance in its entry in the Oxford English Dictionary, which helps to distinguish it from "Arabic numerals" as the Eastern Arabic numerals.
Alternative names are Western Arabic numerals, Western numerals, Hindu–Arabic numerals, Unicode calls them digits. The decimal Hindu–Arabic numeral system with zero was developed in India by around AD 700; the development was gradual, spanning several centuries, but the decisive step was provided by Brahmagupta's formulation of zero as a number in AD 628. The system was revolutionary by including zero in positional notation, thereby limiting the number of individual digits to ten, it is considered an important milestone in the development of mathematics. One may distinguish between this positional system, identical throughout the family, the precise glyphs used to write the numerals, which varied regionally; the first universally accepted inscription containing the use of the 0 glyph in India is first recorded in the 9th century, in an inscription at Gwalior in Central India dated to 870. Numerous Indian documents on copper plates exist, with the same symbol for zero in them, dated back as far as the 6th century AD, but their dates are uncertain.
Inscriptions in Indonesia and Cambodia dating to AD 683 have been found. The numeral system came to be known to the court of Baghdad, where mathematicians such as the Persian Al-Khwarizmi, whose book On the Calculation with Hindu Numerals was written about 825 in Arabic, the Arab mathematician Al-Kindi, who wrote four volumes, On the Use of the Indian Numerals about 830, propagated it in the Arab world, their work was principally responsible for the diffusion of the Indian system of numeration in the Middle East and the West. In the 10th century, Middle-Eastern mathematicians extended the decimal numeral system to include fractions, as recorded in a treatise by Syrian mathematician Abu'l-Hasan al-Uqlidisi in 952–953; the decimal point notation was introduced by Sind ibn Ali, who wrote the earliest treatise on Arabic numerals. A distinctive West Arabic variant of the symbols begins to emerge around the 10th century in the Maghreb and Al-Andalus, which are the direct ancestor of the modern "Arabic numerals" used throughout the world.
Woepecke has proposed that the Western Arabic numerals were in use in Spain before the arrival of the Moors, purportedly received via Alexandria, but this theory is not accepted by scholars. Some popular myths have argued that the original forms of these symbols indicated their numeric value through the number of angles they contained, but no evidence exists of any such origin. In 825 Al-Khwārizmī wrote a treatise in Arabic, On the Calculation with Hindu Numerals, which survives only as the 12th-century Latin translation, Algoritmi de numero Indorum. Algoritmi, the translator's rendition of the author's name, gave rise to the word algorithm; the first mentions of the numerals in the West are found in the Codex Vigilanus of 976. From the 980s, Gerbert of Aurillac used his position to spread knowledge of the numerals in Europe. Gerbert studied in Barcelona in his youth, he was known to have requested mathematical treatises concerning the astrolabe from Lupitus of Barcelona after he had returned to France.
Leonardo Fibonacci, a mathematician born in the Republic of Pisa who had studied in Béjaïa, promoted the Indian numeral system in Europe with his 1202 book Liber Abaci: When my father, appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art soon pleased me above all else and I came to understand it; the numerals are arranged with their lowest value digit to the right, with higher value positions added to the left. This arrangement is the same in Arabic as well as the Indo-European languages; the reason the digits are more known as "Arabic numerals" in Europe and the Americas is that they were introduced to Europe in the 10th century by Arabic-speakers of North Africa, who were using the digits from Libya to Morocco.
Arabs, on the other hand, call the base-10 system "Hindu numerals", referring to their origin in India. This is not to be confused with what the Arabs call the "Hindi numerals", namely the Eastern Arabi
In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.
A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.
Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.
Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a
Latin or Roman script, is a set of graphic signs based on the letters of the classical Latin alphabet. This is derived from a form of the Cumaean Greek version of the Greek alphabet used by the Etruscans. Several Latin-script alphabets exist, which differ in graphemes and phonetic values from the classical Latin alphabet; the Latin script is the basis of the International Phonetic Alphabet and the 26 most widespread letters are the letters contained in the ISO basic Latin alphabet. Latin script is the basis for the largest number of alphabets of any writing system and is the most adopted writing system in the world. Latin script is used as the standard method of writing in most Western, Central, as well as in some Eastern European languages, as well as in many languages in other parts of the world; the script is either called Roman script or Latin script, in reference to its origin in ancient Rome. In the context of transliteration, the term "romanization" or "romanisation" is found. Unicode uses the term "Latin".
The numeral system is called the Roman numeral system. The numbers 1, 2, 3... are Latin/Roman script numbers for the Hindu–Arabic numeral system. The letter ⟨C⟩ was the western form of the Greek gamma, but it was used for the sounds /ɡ/ and /k/ alike under the influence of Etruscan, which might have lacked any voiced plosives. During the 3rd century BC, the letter ⟨Z⟩ – unneeded to write Latin properly – was replaced with the new letter ⟨G⟩, a ⟨C⟩ modified with a small vertical stroke, which took its place in the alphabet. From on, ⟨G⟩ represented the voiced plosive /ɡ/, while ⟨C⟩ was reserved for the voiceless plosive /k/; the letter ⟨K⟩ was used only in a small number of words such as Kalendae interchangeably with ⟨C⟩. After the Roman conquest of Greece in the 1st century BC, Latin adopted the Greek letters ⟨Y⟩ and ⟨Z⟩ to write Greek loanwords, placing them at the end of the alphabet. An attempt by the emperor Claudius to introduce three additional letters, thus it was during the classical Latin period that the Latin alphabet contained 23 letters: The use of the letters I and V for both consonants and vowels proved inconvenient as the Latin alphabet was adapted to Germanic and Romance languages.
W originated as a doubled V used to represent the sound found in Old English as early as the 7th century. It came into common use in the 11th century, replacing the runic Wynn letter, used for the same sound. In the Romance languages, the minuscule form of V was a rounded u. In the case of I, a word-final swash form, j, came to be used for the consonant, with the un-swashed form restricted to vowel use; such conventions were erratic for centuries. J was introduced into English for the consonant in the 17th century, but it was not universally considered a distinct letter in the alphabetic order until the 19th century. By the 1960s, it became apparent to the computer and telecommunications industries in the First World that a non-proprietary method of encoding characters was needed; the International Organization for Standardization encapsulated the Latin alphabet in their standard. To achieve widespread acceptance, this encapsulation was based on popular usage; as the United States held a preeminent position in both industries during the 1960s, the standard was based on the published American Standard Code for Information Interchange, better known as ASCII, which included in the character set the 26 × 2 letters of the English alphabet.
Standards issued by the ISO, for example ISO/IEC 10646, have continued to define the 26 × 2 letters of the English alphabet as the basic Latin alphabet with extensions to handle other letters in other languages. The Latin alphabet spread, along with Latin, from the Italian Peninsula to the lands surrounding the Mediterranean Sea with the expansion of the Roman Empire; the eastern half of the Empire, including Greece, the Levant, Egypt, continued to use Greek as a lingua franca, but Latin was spoken in the western half, as the western Romance languages evolved out of Latin, they continued to use and adapt the Latin alphabet. With the spread of Western Christianity during the Middle Ages, the Latin alphabet was adopted by the peoples of Northern Europe who spoke Celtic languages or Germanic languages or Baltic languages, as well as by the speakers of several Uralic languages, most notably Hungarian and Estonian; the Latin script came into use for writing the West Slavic languages and several South Slavic languages, as the people who spoke them adopted Roman Catholicism.
The speakers of East Slavic languages adopted Cyrillic along with Orthodox Christianity. The Serbian language uses both scripts, with Cyrillic predominating in official communication and Latin elsewhere, as determined by the Law on Official Use of the Language and Alphabet; as late as 1500, the Latin script was limited to the languages spoken in Western and Central Europe. The Orthodox Christian Slavs of Eastern and Southeastern Europe used Cyrillic, the Greek alphabet was in use by Greek-speakers around the eastern Mediterranean; the Arabic script was widespread within Islam, both among Arabs and non-Arab nations like the Iranians, Indonesians, M
String (computer science)
In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and the length changed. A string is considered a data type and is implemented as an array data structure of bytes that stores a sequence of elements characters, using some character encoding. String may denote more general arrays or other sequence data types and structures. Depending on programming language and precise data type used, a variable declared to be a string may either cause storage in memory to be statically allocated for a predetermined maximum length or employ dynamic allocation to allow it to hold a variable number of elements; when a string appears in source code, it is known as a string literal or an anonymous string. In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set called an alphabet. Let Σ be a non-empty finite set of symbols, called the alphabet.
No assumption is made about the nature of the symbols. A string over Σ is any finite sequence of symbols from Σ. For example, if Σ = 01011 is a string over Σ; the length of a string s can be any non-negative integer. The empty string is the unique string over Σ of length 0, is denoted ε or λ; the set of all strings over Σ of length n is denoted Σn. For example, if Σ = Σ2 =. Note that Σ0 = for any alphabet Σ; the set of all strings over Σ of any length is the Kleene closure of Σ and is denoted Σ*. In terms of Σn, Σ ∗ = ⋃ n ∈ N ∪ Σ n For example, if Σ = Σ* =. Although the set Σ* itself is countably infinite, each element of Σ* is a string of finite length. A set of strings over Σ is called a formal language over Σ. For example, if Σ =, the set of strings with an number of zeros, is a formal language over Σ. Concatenation is an important binary operation on Σ*. For any two strings s and t in Σ*, their concatenation is defined as the sequence of symbols in s followed by the sequence of characters in t, is denoted st.
For example, if Σ =, s = bear, t = hug st = bearhug and ts = hugbear. String concatenation is an non-commutative operation; the empty string ε serves as the identity element. Therefore, the set Σ* and the concatenation operation form a monoid, the free monoid generated by Σ. In addition, the length function defines a monoid homomorphism from Σ* to the non-negative integers. A string s is said to be a substring or factor of t if there exist strings u and v such that t = usv; the relation "is a substring of" defines a partial order on Σ*, the least element of, the empty string. A string s is said to be a prefix of t if there exists a string u such that t = su. If u is nonempty, s is said to be a proper prefix of t. Symmetrically, a string s is said to be a suffix of t if there exists a string u such that t = us. If u is nonempty, s is said to be a proper suffix of t. Suffixes and prefixes are substrings of t. Both the relations "is a prefix of" and "is a suffix of" are prefix orders. A string s = uv.
For example, if Σ = the string 0011001 is a rotation of 0100110, where u = 00110 and v = 01. The reverse of a string is a string in reverse order. For example, if s = abc the reverse of s is cba. A string, the reverse of itself is called a palindrome, which includes the empty string and all strings of length 1, it is useful to define an ordering on a set of strings. If the alphabet Σ has a total order one can define a total order on Σ* called lexicographical order. For example, if Σ = and 0 < 1 the lexicographical order on Σ* includes the relationships ε < 0 < 00 < 000 <... < 0001 < 001 < 01 < 010 < 011 < 0110 < 01111 < 1 < 10 < 100 < 101 < 111 < 1111 < 11111... The lexicographical order is total if the alphabetical order is, but isn't well-founded for any nontrivial alphabet if the alphabetical order is. See Shortlex for an alternative string ordering that preserves well-foundedness. A number of additional operations on strings occur in the formal theory; these are given in the article on string operations.
Strings admit the following interpretation as nodes on a graph: Fixed-length strings can be viewed as nodes on a hypercube Variable-length strings can be viewed as nodes on the k-ary tree, where k is the number of symbols in Σ Infinite strings can be viewed as i