Cyrillic alphabet vs alphabet

Cyrillic alphabet and alphabet both are nouns.

Word	Noun	Adjective	Verb	Adverb
Cyrillic alphabet	Yes	No	No	No
alphabet	Yes	No	No	No

As nouns, alphabet is a hypernym of Cyrillic alphabet; that is, alphabet is a word with a broader meaning than Cyrillic alphabet:

Cyrillic alphabet: an alphabet derived from the Greek alphabet and used for writing Slavic languages (Russian, Bulgarian, Serbian, Ukrainian, and some other Slavic languages)
alphabet: a character set that includes letters and is used to write a language

Other hypernyms of Cyrillic alphabet include ABC's, ABC, ABCs, bicameral script.

Cyrillic alphabet (noun)	alphabet (noun)
an alphabet derived from the Greek alphabet and used for writing Slavic languages (Russian, Bulgarian, Serbian, Ukrainian, and some other Slavic languages)	the elementary stages of any subject (usually plural)
	a character set that includes letters and is used to write a language