If possible, please give the English equivalent terms of each and shortly explain what they are.
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You know that Korea is after Josun (roughly 1500 year - 1900 year).
To understand the followings, I suggest that we would know the background. And in further, let me use non-terminology.
유학 is Confucius's study (공자). Confucius is China's scholar. Note that it is Josun's core study.
This study is changed or updated through passing China's countries.
Especially, Confucius's study in Song country period is called Juhee's study (1130-1200), because the scholar Juhee in Song developed it.
-위정척사 : Holding Juhee's study and excluding other studies and religions in end of Josun
-공맹 (사상) : Mencius 맹자 developed Confucius's study. Here their study is called 공맹사상
-가묘 : House displaying ancestor's 위패
-종묘 : House displaying Josun's all kings and queens' 위패 and doing ritual
-위패 : Stick representing the sprit of dead person
-사당 : House displaying family's ancestors' 위패 and doing ritual
-소중화 : 중화 사상 is China's faith, which loves their culture. Here 소중화 is 중화 developed by China's people living in other country.
-원나라 : Monggol, whose territory is China and Monggol mountain, is founded by Kan. 5-th king of Monggol changed the name of country into 원 (roughly 1250 - 1350)
-주자가례 : Juhee's book which tells the etiquette and ritual of noble family called Saedaebu