Differential Operators And Internal State Differential Operators On Semihoops

Hoops were introduced by B.Bosbach in 1969.From the point of view of algebras,hoops are commutative residuated integral monoids.In 2003,semihoop was introduced by Francesc Esteva and Lluis Godo,it has no the separability Therefore,As a more general algebraic structure,semihoops play an important role in the research of fuzzy logic and related algebraic structures.The main contents in this paper are as followsFirstly,we introduce the notions of differential operator and investigate some related properties,define several differential operator and discuss the rela-tionship between them.Secondly,we introduce the concept of differential filters on semihoops and derive some of their characterizations.Then define several different differential filters and study the relationships between them.Then,we discuss the relationships between the set of differential filters and other algebraic structures on semihoops.Moreover,the properties of local differential semihoops are also discussed.Finally,we define the internal state differential semihoops by introducing the internal stated differential operator on semihoops and study the properties of the them.We get the following results(1)The set of all fixed points of a semihoop for an ideal idempotent differ-ential operator is a semihoop(2)Every maximal differential filter is a prime differential filter on ideal differential V-semihoop(3)The set of all differential filters of an ideal differential V-semihoop is a Heyting algebra(4)The equivalent condition for bounded ideal differential semihoops to become local differential semihoops is(ordd)(x(?)y)<? implies(ordd)(x)<?or(ordd)(y)<?.(5)The equivalent condition for idempotent bounded ideal differential semi-hoops to become local differential semihoops is d(x(?)y)=0 implies d(x)=0 or d(y)=0.(6)The set of all principle differential operators of internal state differential semiloops forms a bounded intersection semilattice.