Ideal Theory On Bounded On Semihoops

Semihoops are generalizations of Hoops which were originally introduced by Bosbach named complementary semigroups.Semihoops are the fundamental residuated structures and contain all logic algebras based on residuated lattices.The ideal theory plays an important role of studying logic algebras.In MV-algebras?filters and ideals are dual,but in semihoops?filters and ideals are not dual.The thesis focus on ideal theory on semihoops.Firstly,we introduce the definition of ideals on semihoops?give some equiva-lent characterizations and generation formulas of ideals,and discuss the relations between ideals and filters.Secondly,we give several definitions of special ideals on semihoops,that is,primary ideal,prime ideal,maxiaml ideal and perfect ideal,and study the topological spaces of prime ideal.Finally,we study several different types of semihoops,and study their quotient structure with different ideals.We get the following results:(1)Let A be a bounded V-semihoop with(DNP)and x2=x,for any x?A.Then every primary ideal of A is a prime ideal.(2)Let A be a bounded V-semihoop with(DNP).Then every maximal ideal of A is a prime ideal.(3)Let A be a bounded semihoop.Then every perfect ideal of A is a primary ideal.(4)Let A be a bounded semihoop.Then the topological space of prime ideal is a compact T0 space.(5)Let A be a bounded semihoop with(DNP)and I is a proper ideal of A.Then A/I is local semihoop if and only if I is a primary ideal.(6)Let A be a bounded semihoop with(DNP)and I be a proper ideal of A.Then A/I is finitely local semihoop if and only if I is a maximal ideal.