Derivations and differential filters on semihoops

Abstract

In this paper, we introduce the notions of derivations and differential filters on semihoops and investigate some related properties of them.  Also we discuss related properties of some particular derivations and give some characterizations of ideal derivations.  Then we obtain that the set of all fixed points of a semihoop for an ideal idempotent derivation is a semihoop, the set of all fixed points of a ∨-hoop for an ideal derivation is an ideal.  And for every non-trivial prime ideal I of a semihoop A, there exists a derivation d such that Fix_d(A) = I.  Finally we find some relations between differential filters and other filters.  It is proved that every maximal differential filter is a prime differential filter and the set of all differential filters of an ideal differential ∨-semihoop is a Heyting algebra.