Etymology
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hospitality (n.)
late 14c., "act of being hospitable," from Old French ospitalité "hospitality; hospital," from Latin hospitalitem (nominative hospitalitas) "friendliness to guests," from hospes (genitive hospitis) "guest; host" (see host (n.1)).
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*ghos-ti- 

Proto-Indo-European root meaning "stranger, guest, host," properly "someone with whom one has reciprocal duties of hospitality," representing "a mutual exchange relationship highly important to ancient Indo-European society" [Watkins]. But as strangers are potential enemies as well as guests, the word has a forked path.

The word ghos-ti- was thus the central expression of the guest-host relationship, a mutual exchange relationship highly important to ancient Indo-European society. A guest-friendship was a bond of trust between two people that was accompanied by ritualized gift-giving and created an obligation of mutual hospitality and friendship that, once established, could continue in perpetuity and be renewed years later by the same parties or their descendants. [Calvert Watkins, "American Heritage Dictionary of Indo-European Roots"]

It forms all or part of: Euxine; guest; hospice; hospitable; hospital; hospitality; hospodar; host (n.1) "person who receives guests;" host (n.2) "multitude;" hostage; hostel; hostile; hostility; hostler; hotel; Xenia; xeno-; xenon.

It is the hypothetical source of/evidence for its existence is provided by: Greek xenos "guest, host, stranger;" Latin hostis, in earlier use "a stranger," in classical use "an enemy," hospes "host;" Old Church Slavonic gosti "guest, friend," gospodi "lord, master;" Old English gæst, "chance comer, a stranger."

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xenial (adj.)
"pertaining to hospitality," 1834, from Greek xenia (see Xenia) + -al (1). Related: Xenially.
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hospice (n.)
1818, "rest house for travelers," especially the houses of refuge and shelter kept by monks in the passes of the Alps, from French hospice "hospital, almshouse" (Old French ospice "hospice, shelter," also "hospitality," 13c.), from Latin hospitium "hospitable reception, entertainment; hospitality, bonds of hospitality, relationship of guest and host;" also "place of entertainment, lodging, inn, guest-house," from hospes (genitive hospitis) "guest; host," also "a stranger, foreigner" (see host (n.1)).

Sense of "home for the aged and terminally ill " is from 1879; hospice movement first attested 1978.
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Xenia 
city in Ohio, from Greek xenia "hospitality, rights of a guest, friendly relation with strangers," literally "state of a guest," from xenos "guest" (from PIE root *ghos-ti- "stranger, guest, host"). Founded 1803 and named by vote of a town meeting, on suggestion of the Rev. Robert Armstrong to imply friendliness and hospitality.
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hospitalize (v.)
1873, from hospital + -ize. "Freq[uently] commented on as an unhappy formation" [OED]. As verbs, hospitate is recorded from 1620s as "to lodge or entertain, receive with hospitality" but is rare; hospitize is from 1895. Related: hospitalized; hospitalizing.
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host (n.1)

"person who receives guests," especially for pay, late 13c., from Old French oste, hoste "guest, host, hostess, landlord" (12c., Modern French hôte), from Latin hospitem (nominative hospes) "guest, stranger, sojourner, visitor (hence also 'foreigner')," also "host; one bound by ties of hospitality."

This appears to be from PIE *ghos-pot-, a compound meaning "guest-master" (compare Old Church Slavonic gospodi "lord, master," literally "lord of strangers"), from the roots *ghos-ti- "stranger, guest, host" and *poti- "powerful; lord." The etymological notion is of someone "with whom one has reciprocal duties of hospitality" [Watkins]. The biological sense of "animal or plant having a parasite" is from 1857.

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xeno- 
before vowels, xen-, word-forming element meaning "strange, foreign; stranger, foreigner," from Greek xenos "a guest, stranger, foreigner, refugee, guest-friend, one entitled to hospitality," cognate with Latin hostis, from PIE root *ghos-ti- "stranger, guest, host." "The term was politely used of any one whose name was unknown" [Liddell & Scott].
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hostage (n.)
late 13c., from Old French ostage, hostage "kindness, hospitality; residence, dwelling; rent, tribute; compensation; guarantee, pledge, bail; person given as security or hostage" (11c., Modern French ôtage), which is of uncertain origin. Either from hoste "guest" (see host (n.1)) via notion of "a lodger held by a landlord as security" [Watkins, Barnhart]; or else from Late Latin obsidanus "condition of being held as security," from obses "hostage," from ob- "before" + base of sedere "to sit," with spelling influenced by Latin hostis. [OED, Century Dictionary]. Modern political/terrorism sense is from 1970.
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guest (n.)
Old English gæst, giest (Anglian gest) "an accidental guest, a chance comer, a stranger," from Proto-Germanic *gastiz (source also of Old Frisian jest, Dutch gast, German Gast, Gothic gasts "guest," originally "stranger"), from PIE root *ghos-ti- "stranger, guest, host" (source also of Latin hostis, in earlier use "a stranger," in classical use "an enemy"); the root sense, according to Watkins, probably is "someone with whom one has reciprocal duties of hospitality."

Spelling evolution influenced by Old Norse cognate gestr (the usual sound changes from the Old English word would have yielded Modern English *yest). Meaning "person entertained for pay" (at an inn, etc.) is from late 13c. Old English also had cuma "stranger, guest," literally "a comer." Phrase be my guest in the sense of "go right ahead" first recorded 1955.
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