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Pointed set

In mathematics, a pointed set (also based set or rooted set ) is an ordered pair $$(X, x_0)$$ where $$X$$ is a set and $$x_0$$ is an element of $$X$$ called the base point, also spelled basepoint.

Maps between pointed sets $$(X, x_0)$$ and $$(Y, y_0)$$—called based maps, pointed maps, or point-preserving maps—are functions from $$X$$ to $$Y$$ that map one basepoint to another, i.e. maps $$f \colon X \to Y$$ such that $$f(x_0) = y_0$$. Based maps are usually denoted $f \colon (X, x_0) \to (Y, y_0)$.

Pointed sets are very simple algebraic structures. In the sense of universal algebra, a pointed set is a set $$X$$ together with a single nullary operation $$*: X^0 \to X,$$ which picks out the basepoint. Pointed maps are the homomorphisms of these algebraic structures.

The class of all pointed sets together with the class of all based maps forms a category. Every pointed set can be converted to an ordinary set by forgetting the basepoint (the forgetful functor is faithful), but the reverse is not true. In particular, the empty set cannot be pointed, because it has no element that can be chosen as the basepoint.

Categorical properties
The category of pointed sets and based maps is equivalent to the category of sets and partial functions. The base point serves as a "default value" for those arguments for which the partial function is not defined. One textbook notes that "This formal completion of sets and partial maps by adding 'improper', 'infinite' elements was reinvented many times, in particular, in topology (one-point compactification) and in theoretical computer science." This category is also isomorphic to the coslice category ($$\mathbf{1} \downarrow \mathbf{Set}$$), where $$\mathbf{1}$$ is (a functor that selects) a singleton set, and $$\scriptstyle {\mathbf{Set}}$$ (the identity functor of) the category of sets. This coincides with the algebraic characterization, since the unique map $$\mathbf{1} \to \mathbf{1}$$ extends the commutative triangles defining arrows of the coslice category to form the commutative squares defining homomorphisms of the algebras.

There is a faithful functor from pointed sets to usual sets, but it is not full and these categories are not equivalent.

The category of pointed sets is a pointed category. The pointed singleton sets $$(\{a\}, a)$$ are both initial objects and terminal objects, i.e. they are zero objects. The category of pointed sets and pointed maps has both products and coproducts, but it is not a distributive category. It is also an example of a category where $$0 \times A$$ is not isomorphic to $$0$$.

Applications
Many algebraic structures rely on a distinguished point. For example, groups are pointed sets by choosing the identity element as the basepoint, so that group homomorphisms are point-preserving maps. This observation can be restated in category theoretic terms as the existence of a forgetful functor from groups to pointed sets.

A pointed set may be seen as a pointed space under the discrete topology or as a vector space over the field with one element.

As "rooted set" the notion naturally appears in the study of antimatroids and transportation polytopes.

title: Pointed set


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The Fox (novella)

The Fox is a novella by D. H. Lawrence which first appeared in The Dial in 1922. Set in Berkshire, England, during World War I, The Fox, like many of D. H. Lawrence's other major works, deals with the psychological relationships of three protagonists in a triangle of love and hatred. Without the help of any male laborers, Nellie March and Jill Banford struggle to maintain a marginal livelihood at the Bailey Farm. A fox has raged through the poultry, and although the women—particularly the more masculine Nellie—have tried to shoot the intruder, he seems always to elude traps or gunshot.

Plot introduction
Banford and March live on a farm together because it does not look like they will marry. Although they are only in their late twenties, in that era women who were still single at their age were generally considered to have forgone the prospect of marriage. Banford is thin and frail, in contrast to her companion who is physically masculine. However particular emphasis is given to March's face, which is feminine and expressive. The women are depicted as fearful of femininity and fertility. For example, they sell a heifer before it calves.

The fox becomes a hindrance to Banford and March, but March finds she cannot hunt it, and rather, she becomes entranced by it. Shortly after this, Henry, a young man, comes to stay with the women, and a link is established between the fox and Henry. This intriguing novella explores gender roles, sexuality, femininity, and the pity of war, as do two other Lawrence novellas written at the same time, The Ladybird and The Captain's Doll.

Film adaptation
A 1967 film was made from The Fox, starring Sandy Dennis as Jill Banford, Anne Heywood as Ellen March, and Keir Dullea as Paul (not Henry) Grenfield.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: The Fox (novella)


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Theophoric name

A theophoric name (from Greek: θεόφορος, theophoros, literally "bearing or carrying a god") embeds the word equivalent of 'god' or God's name in a person's name, reflecting something about the character of the person so named in relation to that deity. For example, names embedding Apollo, such as Apollonios or Apollodorus, existed in Greek antiquity.

Theophoric personal names, containing the name of a god in whose care the individual is entrusted (or a generic word for god), were also exceedingly common in the ancient Near East and Mesopotamia. Some names of theophoric origin remain common today, such as Theodore (theo-, "god"; -dore, origin of word compound in Greek: doron, "gift"; hence "God's gift"; in Greek: Theodoros) or less recognisably as Jonathan (from Hebrew Yonatan/Yehonatan, meaning "Yahweh has given").

Classical Greek and Roman theophoric names
Certain names of classical gods are sometimes given as personal names. The most common is Diana and its variants, such as Diane; others include Minerva, Aphrodite, Venus, Isis, or Juno. The first pope to take a regnal name, Pope John II, had the given name Mercurius and changed his name as he considered it inappropriate for the Pope to have the name of a pagan deity.

Christian theophoric names

 * Abdelmasih: (Arabic) "servant of the Messiah"
 * Abdisho: (Syriac) "servant of Jesus"
 * Ahadabui, also Ahidabu: (Syriac) "brother to the Father"
 * Ahischema: (Syriac) "brother of the schema"
 * Aitillaha: (Syriac) "God exists"
 * Amadeus: (Latin) "lover of God"
 * Attallah: (Arabic) "gift of God"
 * Bakhtishu: (Syriac) "redeemed by Jesus"
 * Bogdan: (Slavic) "God given"
 * Bogomil: (Slavic) "dear to God"
 * Bozhidar: (Slavic) "gift of God"
 * Christian: (Greek) "believer in Christ"
 * Christodoulos: (Greek) "servant of Christ"
 * Christopher: (Greek) "Christ-bearer".
 * Daniel: (Hebrew) "God is my Judge"
 * Deodatus/Deusdedit: (Latin) "God-given"
 * Dorotheus/Dorothea: (Greek) "gift to God"
 * Fürchtgott: (Germanic) "God-fearing"
 * Gebreamlak: (Ge'ez) "servant of God"
 * Geoffrey/Gottfried: (Germanic) "God's peace"
 * Gottlieb: (Germanic) "God's love"
 * Ishodad: (Syriac) "given by Jesus"
 * Ishosabran or Sabrisho: (Syriac) "patient for Jesus"
 * Ishoyahb or Yahbisho: (Syriac) "Jesus has given"
 * Jonathan: (Hebrew) "God has given"
 * Marnazka: (Syriac) "the Lord has conquered"
 * Michael: (Hebrew) "who is like God"
 * Philothea/Philothei/Philotheos: (Greek) "lover of God"
 * Sabrisho or Ishosabran: (Syriac) "patient for Jesus"
 * Shubhalisho, also Shubhisho: (Syriac) "praise to Jesus"
 * Shenouda: (Coptic) "son of God"
 * Slibazka: (Syriac) "the Cross has conquered"
 * Raphael: (Hebrew) "God has healed"
 * Theodore/Theodora: (Greek) "gift of God"
 * Theodosius/Theodosia, Theodotos/Theodotē and Dositheus/Dosithea: (Greek) "God-given"
 * Theodotus: (Greek): "given by God"
 * Theodulus: (Greek) "servant of God"
 * Theophilus: (Greek) "one who loves God"
 * Theognis: (Greek) "God-knowing"
 * Theophanes/Theophania, Tiffany: (Greek) "manifestation of God"
 * Theophrastus: (Greek) "godly speech"
 * Theaetetus: (Greek) "one who pleads to God"
 * Timothy/Timotheus: (Greek) "one who honors God"
 * Yahballaha: (Syriac) "God has given"

Some Christian saints have polytheistic theophoric names (such as Saint Dionysius, Saint Mercurius, Saint Saturninus, Saint Hermes, Saint Martin of Tours, Saint Demetrius of Thessaloniki).

Germanic theophoric names

 * Os, meaning "god"
 * Oslac
 * Oswald
 * Oswin
 * Thor, the god of thunder
 * Thorstein means "Thor's stone"
 * Thorkel means "Thor's craft"
 * Thorulf means "Thor's wolf"
 * Thordis
 * Thora
 * Ing, an old name for Freyr (an epithet meaning "lord")
 * Ingrid
 * Ingeborg
 * Inger
 * Ingunn

Rarely, Germanic names contain the element Wod (such as Woðu-riðe), potentially pointing to an association with the god Odin. In connection, numerous names containing wulf "wolf" have been taken as totemistic, expressing association with Odin in the earliest period, although -ulf degenerated into a mere suffix from an early time (Förstemann 1856).

Hinduism
The personal names of almost all gods and goddesses of various deities from the polytheistic Hindu pantheon are considered common and traditional names for people from region. Many traditional Hindu names are in fact from various names or epithets of Hindu gods or goddesses. This is in addition to compound theophoric names using the name of a deity in addition to possessive qualifiers.

 * Names of gods which are also used as personal names, include
 * Vishnu
 * Shiva
 * Lakshmi
 * Parvati
 * Indra
 * Personal names using a deity's name as the base
 * Vaishnavi, meaning "a worshipper of Vishnu"
 * Shivansh, meaning "a part of Shiva"

Brahma, the Hindu creator god, is one of the only deities of the pantheon whose name is rarely if ever used as a personal name or as a base for theophoric personal names.

Some seemingly theophoric names, may in fact be more related to the original etymology of the deity's name itself. For example, both Lakshmi (fortune, success, prosperity) and Lakshman (prosperous, principal, marked) are names of a deity and an avatar respectively, which are related to lakṣ meaning "to mark or see".

Islam

 * Abdullah: "servant of God"

Judaism and biblical
Much Hebrew theophory occurs in the Bible, particularly in the Old Testament. The most prominent theophory involves
 * names referring to El, a word meaning might, power and (a) god in general, and hence in Judaism, God and among the Canaanites the name of the god who was the father of Baal.
 * names referring to Yah, a shortened form of Yahweh.
 * names referring to Levantine deities (especially the storm god, Hadad) by the epithet Baal, meaning lord.

In later times, as the conflict between Yahwism and the more popular pagan practices became increasingly intense, these names were censored and Baal was replaced with Bosheth, meaning shameful one. However the name Yahweh does not appear in theophoric names until the time of Joshua, and for the most part is very rare until the time of King Saul, when it began to be very popular.

El

 * Abdiel: "Servant of God"
 * Ariel: "lion of God"
 * Daniel: "God is my judge" or "justice from God"
 * Elijah: "my God is YHWH"
 * Elihu: "He is my God"
 * Elisha: "my God is salvation"
 * Elisheba (Elizabeth): "my God is an oath" or "my God is abundance"''
 * Immanuel: "God is with us"
 * Yehezkel (Ezekiel): "God will strengthen"
 * Gabriel: "man of God" or "power of God"
 * Ishmael: "God listens"
 * Israel: "who struggles with God"
 * Joel: "YHWH is God"
 * Lemuel: "Dedicated/Devoted to God"
 * Michael: "Who is like God?"
 * Nathaniel: "God-given" or "gift of God"
 * Raphael: "God heals/God is great"
 * Samuel: "God heard"
 * Uriel: "God is my light"
 * Uzziel: "God is my strength"

Yahweh
The name of the Israelite deity YHWH (usually shortened to Yah or Yahu, and Yeho or Yo) appears as a prefix or suffix in many theophoric names of the First Temple Period. For example, Yirme-yahu (Jeremiah), Yesha-yahu (Isaiah), Netan-yah, Yedid-yah, Adoni-yah, Nekhem-yah, Yeho-natan (Jonathan), Yeho-chanan (John), Yeho-shua (Joshua), Yeho-tzedek, Zekharya (Zechariah).

"Yahū" or "Yah" is the abbreviation of YHWH when used as a suffix in Hebrew names; as a prefix it appears as "Yehō-", or "Yo". It was formerly thought to be abbreviated from the Masoretic pronunciation "Yehovah". There is an opinion that, as Yahweh is likely an imperfective verb form, "Yahu" is its corresponding preterite or jussive short form: compare yiŝtahaweh (imperfective), yiŝtáhû (preterit or jussive short form) = "do obeisance".

 * Abijah: "my father is YHWH"
 * Adonijah: "YHWH is the Lord"
 * Hezekiah: "YHWH strengthens"
 * Isaiah: "YHWH is salvation"
 * Jedediah: "friend of YHWH"
 * Jeremiah (Jeremy): "YHWH will raise"
 * Jonathan: "YHWH has given"
 * Joseph: "YHWH shall increase"
 * Josiah: "YHWH saves"
 * Matityahu (Matthew): "gift of YHWH"
 * Micah/Micaiah: "who is like YHWH?"
 * Nehemiah: "YHWH comforts"
 * Obadiah: "servant of YHWH"
 * Toviyahu (Tobias): "the goodness of YHWH"
 * Uriah: "YHWH is my light"
 * Uzziah: "YHWH is my strength"
 * Yehoshua (Joshua)/Yeshua (Jesus): "YHWH will save"
 * Yohanan (John): "graced by YHWH"
 * Zechariah (Zachary): "YHWH has remembered"
 * Zephaniah: "hidden by YHWH"

In the table below, 13 theophoric names with "Yeho" have corresponding forms where the letters eh have been omitted. There is a theory by Christian Ginsburg that this is due to Hebrew scribes omitting the "h", changing Jeho into Jo, to make the start of "Yeho-" names not sound like an attempt to pronounce the Divine Name.

Referring to other gods

 * Jerubbaal, the alternate name of Gideon, variously translated as "Baal will contend"
 * Jezebel: "glory to Baal"
 * Ishbaal: "man of Baal"
 * Balthazar and Belshazzar (Babylonian): "Baal, protect the king"
 * Abijam: "my father is Yam"
 * Shalmaneser (Assyrian): "Shulmanu is foremost"
 * Sennacherib (Assyrian): "Sîn has replaced the brothers"
 * Pygmalion (Phoenician via Greek): "Pummay has given"
 * Nebuchadnezzar (in Babylonian: Nabu-kudurri-usur): "Nabu, watch over my heir"
 * Mordecai: "from Marduk"
 * Ben-Hadad: in Hebrew means "son of Hadad", but his original Aramaic name is Hadadezer: "Hadad is help"
 * Mark: "dedicated to Mars"

Theophoric names containing "Baal" were sometimes "censored" as -bosheth = "shameful one", whence Ishbosheth etc.

title: Theophoric name