As question suggests: what does き stands for in 招き猫? Is it some sort of particle, that "connects" 招(invitation) and 猫(cat) or something else?
1 Answer
No, き here is a part of the 招き - invitation. It's an okurigana, you can say that it's a reading hint to know that 招 should be read using a kun reading - まねき and not an on reading - しょう.
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As far as I'm aware, the on-reading for 猫 is extremely rare. The only example of such a kanji compound I could find in Jisho, for example, is 猫額 "cat's forehead", used as a metaphor for "small thing". And this is one of the cases where the on-reading is very different from the modern Chinese pronunciation, too. From what I can tell with Wiktionary, the /b/ consonant sound is particular to some Min Nan (especially Hokkien) dialects, one of which seems to have been picked up somewhere in the Tang dynasty (since it's listed as a kan-on reading). So, no, not realistically. Nov 19 at 20:17
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Aside from that, without any other context 「しょうびょう」 would almost certainly be 「傷病」 - that's basically all my IME will suggest. Nov 19 at 20:19
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猫額 "cat's forehead", used as a metaphor for "small thing"
-- We use it(usually [猫]{ねこ}の[額]{ひたい}) to talk about 狭い土地 rather than "small thing"– chocolate ♦Nov 19 at 23:58