public school vs lyceum

public school and lyceum both are nouns.

Word	Noun	Adjective	Verb	Adverb
public school	Yes	No	No	No
lyceum	Yes	No	No	No

As nouns, lyceum is a hypernym of public school; that is, lyceum is a word with a broader meaning than public school:

public school: private independent secondary school in Great Britain supported by endowment and tuition
lyceum: a school for students intermediate between elementary school and college; usually grades 9 to 12

Other hypernyms of public school include Gymnasium, lycee, middle school, secondary school.

public school (noun)	lyceum (noun)
a tuition free school in the United States supported by taxes and controlled by a school board	a public hall for lectures and concerts
private independent secondary school in Great Britain supported by endowment and tuition	a school for students intermediate between elementary school and college; usually grades 9 to 12