On derivations of commutative multiplicative semilattices.

In this paper, we investigate related properties of some particular derivations and give some characterizations of regular derivations in commutative multiplicative semilattices. Also we give some characterizations of zero derivations in prime commutative multiplicative semilattices. Then we prove that the set of all prefect derivations ideals on commutative multiplicative semilattices with prefect derivations can form a complete Heyting algebra and obtain that there exists a one to one correspondence between the set of all prefect derivations ideals on commutative multiplicative semilattices with prefect derivations and its quotient structure. Finally, we show that the structure of an idempotent commutative quantale is completely determined by its set of all principal derivations. [ABSTRACT FROM AUTHOR]